Hello! Welcome back to the world of Hexcells with its first sequel, Hexcells Plus. This is our brand-new, 100% no-mistake walkthrough. As with the previous guide, this one will attempt to break down all of the new puzzles to work out the solutions logically, without guessing.

One word of caution before continuing, however. Whereas the original Hexcells took its time with introducing new concepts and easing you into more difficult puzzles, Hexcells Plus wastes very little time in introducing more difficult concepts. After a brief refresher course on the basics, you'll quickly be thrown into some very tricky situations!

Hexcells Plus assumes that you have played through the original. If you for some reason are starting with this game, do yourself a favor: Either read through some of the puzzles in my previous guide, or go back and play the original first. Otherwise, you may be quickly overwhelmed. If you need to read/re-read my previous guide, do so here:

https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=320523083

The controls are identical to the original. Use the Left Mouse Button to mark a cell you feel is part of the pattern, or to highlight a line of cells. Use the Right Mouse Button to eliminate cells you feel are not part of the pattern, and to dim out the headers for those lines you feel are completed. If you make a mistake, don't worry; a second right-click will embolden the line header again.

Later on, you'll deal with blue hexes with a number inside of them. Left-clicking on one of these will expand a grid--an overlay--out across the puzzle; we'll talk about how to use this mechanic in plenty of detail when the time comes, but know that left-clicking on such a cell either expands or retracts the highlighted grid associated with it, while right-clicking on it dims the number within if you believe you are done with it. Again, if you dim one out too early, a second right-click will restore the number. Don't worry about applying this yet; just understand the basics of controlling them. There will be dedicated strategies for handling these in their respective puzzles.


During the opening puzzles in Hexcells Plus, we will revisit the concepts we learned in the original; think of this section as an appetizer before we begin the main course. Here's what you need to know to get started:

• The overall goal is to either "color" (or "mark"; the terms are interchangeable) the orange hexes on the grid blue (by left-clicking), or to eliminate them from the game (by right-clicking).
  
  
• The black hexes on the grid are "empty" hexes. The number inside one lets you know how many of its adjacent cells will be colored blue. Some, marked with a "0," will have no blue hexes around them; some may have all of their neighboring hexes marked.
  
  
• Orange hexes which are eliminated will become empty hexes; they will typically provide clues on how to solve their neighboring empty hexes.
  
  
• The concept of "shared hexes" is vital. In general, it refers to any hex which can contribute to the requirements of more than one entity on the board, whether that's multiple empty hexes, multiple lines of hexes, and so on. The term most often applies to blue hexes, such as a blue hex shared among a group of surrounding empty hexes, but it can also apply to any hex that is involved in the solution of two or more restrictions. Specific applications of the term will be explained in detail throughout the guide.
  
  
• When dealing with a line of cells, click on the numbered header to draw a line through all cells considered to be governed by that number. Be sure to consider the line restrictions in addition to the empty cells on the grid; you have to work with both to accurately determine which cells to mark or eliminate.

This is everything you need to know to get started; the rest will be learned during the walkthrough. As before, the game is divided into six chapters; this time, however, each chapter has six puzzles apiece, for a total of 36 puzzles. As you complete each one, you will earn a certain number of hexes, which will add together to unlock each new chapter. In the puzzle index, you'll see the number of hexes still remaining to be collected to unlock each chapter. As before, any mistake made subtracts from the number you will earn after you complete the puzzle.

At the top-right corner of the puzzle board, you will see two counters. The "REMAINING" counter shows the number of blue hexes yet to be marked in the current puzzle; the "MISTAKES" counter simply shows how many mistakes you have made so far in solving it. At the end of a puzzle, you can either advance to the next one, retry the current puzzle, or return to the puzzle index. Here's a shot of the first puzzle to show the basics:



If you pay attention to the annotations, you've got a couple of hints towards starting that first puzzle. ;-) Anyway, that's enough for an introduction; it's time to get to the meat and potatoes of the game! Let's go ahead and begin Chapter 1 of Hexcells Plus!

The introductory puzzle reintroduces us to the basic concepts of Hexcells but is not so simplistic as to insult anyone who has gone through the original game. We get one reasonably large grid with a few empty hexes to start us out. Here's the layout:



The cardinal rule of Hexcells is to always start with the most obvious moves first. Let's examine the empty cells. We actually get two zeroes, which helps us out immensely. We know that a "0" has everything around it erased, so this allows us to erase seven cells immediately.

In the top-center, we have a "3", which means that three of its adjacent cells will be blue. Only three total orange hexes surround the "3", meaning they will all be marked. So our opening move looks like this:





You may have noticed that a couple more zeroes opened up near the bottom; though it only eliminates two more, we can go ahead and clear the cells bordering them. Now, notice that empty "1" cell we revealed in the very center. Remember the concept of shared hexes from the introduction? Here's a good example. Notice how the blue cell above the "1" is right in between both it and the "3" at the top. This means that blue hex is "shared" by both of them. The "1", however, still has two orange hexes bordering it; since we have a blue hex shared between it and the "3", we can clear these two orange hexes, leaving us with this:

So far, so good. There are actually a few moves we can make now. We revealed a "4" towards the right with the last cells we eliminated; you might see that there are only four "active" cells surrounding it. (I'll define "active cells" momentarily.) With two marked blue and only two orange cells remaining around it, we can safely mark both of them.

You might now notice that the blue cell we just marked on the bottom-right corner of the "4" just gave a blue hex to the "1" right below it--another example of a shared blue hex. Now, just clear the two orange hexes still touching that "1". Interesting; we just revealed another "0". If we clear the single hex bordering it, we see how to solve this side of the puzzle:


Before continuing, I do want to address the concept of "active hexes." We know that the black cells are called "empty hexes." They have a final solution, and we know they are not part of the pattern. I consider both orange and blue hexes to be "active hexes." Orange hexes are considered active simply because we have taken no action on them; they have yet to be solved one way or the other. Blue hexes are considered to be active because they are a part of the pattern. This is another concept that will be revisited throughout this guide.

Returning now to the puzzle, the left side is just a little trickier, but not by much if you take your time. We can approach it either from the "3" just to the left of the center, or from the "2" at the far left edge beneath the "0"; either way will get you to the solution. I'm going to proceed from the "2" on the left and work back towards the center. If you look carefully, you can see that there are only two remaining active hexes around it; the other two are empty hexes. This tells us that both must be colored.

This leaves us an interesting pattern to work with. The "REMAINING" counter shows us that only one of the final orange hexes can be marked blue. To determine which one, notice that we have an empty "1" cell in a cluster of six hexes, five of which are empty. By marking its lone orange hex, we give a blue hex to it, the "3" above, and both of the nearby empty "2" cells. Since this completes all of the blue hex requirements for the puzzle, erase the final two cells to complete it.



Hexes Earned for Completing This Puzzle: 3

We continue the Hexcells Plus refresher course in Puzzle 1-2. Here's the layout:



It's a lot more straightforward to start with the bottom section on this one, though you could start with the top if you wanted to. The logic up here is a little trickier, so for this walkthrough, I'm going to start with the bottom, which has a couple of zeroes to get things started. Go ahead and clear the hexes around them. What we're left with, then, is a pair of empty "2" cells sharing only two orange hexes between them. So we just need to mark both to solve each "2". Notice also that the upper blue cell is also shared by the empty "1" cells we just revealed over top of each "2", so we can immediately clear the three orange hexes bordering them. So we start with this:

Notice that our opening moves have already solved some 40% of the puzzle! The next step is only slightly trickier. Look across the top row of that bottom section; we've now got both a "2" and a "1" with only two orange hexes apiece. The "2" claims both of its cells, one of which is shared with the "1", so its second orange hex will be erased.

The final step in the section builds on the relationships we just established. On the left side, the "1" right above the "0" just got its blue hex from the last sequence; when we clear the cell remaining on its top-left edge, we leave only one remaining orange hex for the "1" in that corner to claim--which, of course, also completes the "2" we just revealed. Finally, over on the right side, those two remaining orange hexes both go to the "2" right under them; after all, there are no additional hexes left for it to claim. Marking them also gives a shared blue hex to both the "1" on the upper-left edge of the "2", as well as the "1" positioned above the first "1". So now, clear the cell directly above that upper "1".

We're in great shape to start the top section now. The "1" we just revealed working our way up the right-hand column has only a single hex above it to claim. We'll go ahead and mark it, but how do we proceed from here? We don't have any nearby empty cells or line headers to tell us where to go next.

We have to move next to that triangle of empty cells at the top of the grid. We get a "2" at the very top, and two empty "1" cells below that. The "2" has only three adjacent orange cells in a ring around the bottom. Notice that each of those cells is shared by at least one of the empty "1" cells below.

What this tells us, then, is that whichever cells the "2" claims will also complete the empty "1" cells. It also tells us that the hexes directly above each "1" must be colored; if we were to color the center cell in that trio, it would be impossible to give a second blue hex to the "2" without also giving a second one to at least one of the empty "1" cells. So the next sequence looks like this, after clearing out all of the remaining hexes from the pair of empty "1" cells:

The rest of the puzzle is actually pretty straightforward; we've just completed the hardest part! Let's work from right to left. If you see, the two orange hexes in the next-to-last column each border a "1" which already has a blue hex. So we simply eliminate both of them. From the pair of empty cells this reveals, we can easily see that the final cell on that side must be marked.

If we now follow the bottom row of cells in this section back to the left, we have a virtually identical situation to what we had along the top row of the bottom section. The "2" will claim the remaining hex next to it, also giving one to the "1" next-door; clear the orange hex along its bottom-left edge. We can't quite solve the "1" that this reveals yet, though.

Let's finish this puzzle. We've got another column of two orange hexes like we had over on the right; the top one again borders a "1" which already has a blue hex. When we clear that one, we can see immediately that the "2" this reveals will claim both orange hexes in that corner. Guess what? There are no blue hexes remaining to be marked! Clear the final cell, and let's get out of here!



Hexes Earned for Completing This Puzzle: 4

We actually get a small difficulty spike starting with Puzzle 1-3. Whereas the last two puzzles gave us fairly straighforward relationships between the cells to work through, this one actually gives us a few minor brain-teasers to pore over.




We get six empty hexes to start us off. The only one we can solve right off the bat, though, is the "3" in the top-left corner. Since it has only three orange hexes rimming the bottom, they will be marked. One of them is shared by the "1" over towards the left edge of the cluster, so the remaining two hexes around it will be cleared. The "0" revealed here is of no real assistance, but the "2" on the other hand is in a position to claim two of the blue hexes we just marked. When we clear the orange hex on its bottom-right edge, we see that the "1" right below it now has only one hex it can claim. Marking it also solves the "2" we just revealed, and the final two hexes of this section are then cleared.

So let's continue into the central section which connects the three larger clusters. We can immediately see that the "1" here already has a blue hex. The next cell over is erased, which gives us another "1". A roadblock; with three possible cells to mark, it's impossible to solve it immediately.

Let's move over to the right side and see if we can find a way to join back up to the middle. We again have a "3" and a "1", though this time, the choices for their blue hexes are much less obvious. So let's examine what we do know. The "3" has a ring of four orange hexes surrounding it. Notice that of those four hexes, the top two are also connected to the "1". We know that three of the four hexes in this ring must be colored. Because it's impossible to give the "3" a complete set of blue hexes without also giving one to the "1", we know that the "1" cannot possibly claim the hex on its bottom-left edge. It also guarantees that the hexes along the bottom-left and bottom-center edges of the "3" are blue. Here's a closer look to show what I mean:



So when we eliminate the hex on the bottom-left edge of the "1", we get a "0", which blows this part of the puzzle wide open. The ensuing sequence even reveals a few more zeroes, eliminating most of the remaining cells in the cluster; in fact, it turns out that the only blue cells we get here are the ones surrounding the "3" that we started with!



Sadly, the "1" we revealed closer to the central column at the end of that sequence is in the position where two hexes are available, without any way to make an obvious selection to clear the section. So we'll need to tackle the bottom next.

We again have a "3" and a "1", and if you look at the positioning, there's a very similar logic here. The "3" again has a ring of four cells around it, two of which again are shared by the "1". So we know which two hexes are guaranteed to go to the "3", and which hexes are guaranteed not to go to the "1"!



Unlike before, though, we can't immediately solve the "3" and the "1", so we'll have to work just a little harder. Notice, however, that the "1" we revealed near the bottom-right corner has only one choice for a blue hex; marking it gives a shared blue hex to the "1" directly above it, letting us clear its two remaining orange hexes. We get a "0" and a "2" from this; the next sequence is very straightforward:

The center column leaves us with another brain-teaser; even with having solved some 95% of the puzzle, we still don't have an obvious way to determine which hexes those empty "1" cells in the middle will claim. We need to use the "REMAINING" counter to help us here. We have four orange hexes, and two still need to be marked. That's a very important clue! Look at the second orange hex down from the top in this group; see how it's shared between both of the empty "1" cells? This means it cannot be marked! Why? Because it would actually force us to erase all of the puzzle's remaining hexes, also preventing the "1" on the top-left edge of the "0" down below from even claiming a blue hex.

So the orange hex shared between the empty "1" cells at the top of the central column can now be erased, which reveals yet another "1". This erasure leaves the "1" on its top-right edge with only one final hex it can claim, which, of course, will be shared with both of them. We now know that the hex below the "1" we just revealed has to be cleared; mark the final cell to complete the puzzle.



Hexes Earned for Completing This Puzzle: 5

Puzzle 1-4 reintroduces the concept of marking consecutive blue hexes around an empty cell. For example, an empty cell marked {3} not only has three blue hexes around it, but they all must be consecutive. Here's the layout:




Special Note: The game formally identifies the symbols as "brackets," but in mathematics, they are actually called "braces." Therefore, that is what I call them throughout these walkthroughs. If you prefer the in-game notation, just mentally switch the words; there is absolutely no impact on gameplay, just a difference in terminology. Real brackets, by the way, are these: [ and ].

So we have one very small grid on the left and one very large grid to make up the rest of the puzzle. Let's start with the small one. We get a {3} near the top, and a {2} near the bottom. One thing that we always know: If a {2} has only three surrounding orange hexes, and they are all contiguous--this is important!--the center hex will always be marked. Similarly, if we have a {3} with only four contiguous surrounding hexes, the center two will always be marked.

We're going to apply these principles to solve this puzzle. Let's start with the {2}; if we apply the rule we just discussed and grant a blue hex to the center cell around it, we will also start the chain of hexes around the {3}. Because that hex is both shared with the {3} and also represents an endpoint in surrounding ring of hexes, we just need to mark the next two cells in succession going around it--consequently solving both empty hexes at once. Clear the final two hexes in the section.

Moving to the bottom of the main grid now, we're going to put the second rule into play. We again get a {3} near a {2}, only this time, the {2} has four hexes around it; there's no possible way to determine which will be marked yet. However, the {3} also has four hexes around it. Remember: The center two must be marked. Why does this work? Well, try mentally numbering the cells in order. Let's number the cell on its top-left edge "1" and the one on its bottom-right edge "4." In order to get a continuous chain of three blue hexes, there are only two combinations: Cells 1, 2, and 3, or Cells 2, 3, and 4. Notice that Cells 2 and 3 are included in both combinations; thus, they are guaranteed blue hexes!

So let's go ahead and mark these two cells. We have now fed a blue hex to the {2} within the middle of its own cluster. We may not know which other cell it will claim yet, but we can now see which one it cannot claim--the one at the very edge of the chain, on its top-right edge. Clear that cell; we get a "0", setting up two eliminations and telling us how to solve both the {3} and the {2}.


If you went ahead and cleared the cells around the second "0" just revealed, that's fine; you can always do this as soon as you see them. I proceeded this way just to put an emphasis on the logic of this section. We are going to go ahead and clear those hexes now. Doing so sets us up to complete the right side of the grid. Notice that the new {3} we just revealed has only three orange hexes to start with. Naturally, they will all be marked. This feeds a hex to the {2} at the right edge; notice that it's at an endpoint in its chain of three active cells. Mark the next one up, thereby also giving a second consecutive blue hex to the {3} across from it. Go ahead and mark the next consecutive hex in the chain around this {3}, then clear the last two cells in the cluster.

We now just need to finish the central section. Notice that the "1" on the top-left edge of the {3} shares a blue hex; go ahead and clear the cell right above the "1". Another "1"; we'll actually keep following this as we ultimately unveil a chain of empty "1" cells which all share this same hex. Heading up the central column, we get a "0"; follow it, too. A {3}, again with only three hexes to deal with. So now, we have this:

This leaves only the {2} cells in the corners to solve, and each has a blue hex from that last sequence. We'll solve them as before: By picking out the cells they cannot possibly claim. The empty hexes that this reveals will guide us to the solution; can you figure it out?



Hexes Earned for Completing This Puzzle: 5

Whereas Puzzle 1-4 reintroduced the concept of marking connected groups of cells around an empty hex, Puzzle 1-5 reintroduces the idea of marking blue hexes that are not connected. For example, take a negative integer such as -3-; we still have three blue hexes surrounding such a cell, but at least one of them will have an empty hex separating it from the others. We'll describe this in much more depth as we continue the walkthrough.




So there are a couple of assumptions we can make regarding negative empty hexes. Let's start with the -3- just left of the center. It has four orange hexes surrounding it. If we number them similarly to how we numbered cells in the last puzzle, we can determine that the first and fourth hexes surrounding it must be marked. Why? Because there are only two patterns that will net you three non-consecutive blue hexes: Cells 1, 2, and 4, and Cells 1, 3, and 4. So we can go ahead and mark the hexes directly above and below the -3-.

Note that this gave a blue hex to the -2- just down from the -3-; the cells around it are spaced such that there are two active hexes on its right, and only one active hex on its left. Since we need a second, non-consecutive blue hex, we need to clear the cell below the one we just marked, then mark the lone hex on the left side of the -2-. So our opening is this:


Notice that we have a second -2- on the bottom-left edge of the first one; the blue hex we just marked is thus shared with it. The hex on the top-left edge of this -2- will immediately be cleared, since marking it would violate the non-consecutive rule governing the -2-. We get a -3- from this; notice that it is positioned to have only three total active hexes surrounding it, with one already blue. Marking the others will solve it, also giving a second blue hex to the -2- and letting us clear the final orange hex directly below it.

This elimination gives us a "1", which we can immediately see shares the blue hex we just marked. Clearing the orange hex on its bottom-left edge gives us a "2". We see immediately that it will claim the final orange hex in the section.





We now want to move to the -2- at the bottom-center of the grid. We've already given it a blue hex; since the next one can't be connected to this one, erase the next cell over in its ring of hexes. A "2" is uncovered, but it more importantly eliminates one of the remaining active hexes surrounding the -3- on its upper-left edge. We can now mark the remaining hex on the upper-right edge of the -3- to solve it. Doing so also gives a second blue hex to the "2" we just revealed. Clearing the remaining orange cell on the bottom-right edge of the "2" then leaves only one remaining hex for the -2- to claim.

Continuing onward to the right, just clear the orange hex adjacent to the "1" we just revealed. Another -2-; we'll follow the same basic pattern, clearing the next hex in sequence from the blue one it already has. The "2" revealed has only one choice for a second blue hex; marking it also gives the -2- the second one it needs, and we can clear the final hex in its cluster. The "1" that this reveals will allow us to solve the -4- in this corner:

We only have the top section left to solve. Solving the -2- near the right-center is very straightforward; we now have only three active hexes around it now. The first one is already marked; erase the center hex and mark the third one in line. Notice that the "2" we just revealed shares those same two blue hexes; erase the orange hex along its top-left edge.

Special Note: The pattern of a -2- surrounded by exactly three continuous active hexes is one you should learn to recognize. Any time you see this, you will mark the endpoints and erase the center hex every single time! Empty hexes around such a -2- are fine as long as they do not break the chain of three continuous active hexes.

This elimination gives us another "2", which we immediately see has two blue hexes. Clear the hex above it to reach the top cluster. Another -2-. Don't let this cluster of -2- cells trick you! The one we just revealed has only three active hexes around its top edge. Solve it the same way: Mark the first and third cells in the ring, then clear the middle one. When you do this, you will reveal a "3" right in the center of this section; it will tell you exactly how to complete the puzzle.



Hexes Earned for Completing This Puzzle: 5

Chapter 2 unlocks at this point; as always, you can check out the next series of puzzles in advance if you would like. We complete Chapter 1 with a test over our knowledge of sequential and non-sequential blue hexes.


It sort of looks like a Magic Mushroom from old-school Super Mario Bros....



Most of the empty cells we're given are negative integers. We do get a {2} in the top-left and a "1" near the right-center, but the rest are going to have disconnected groups of blue cells around them. Notice that the -2- at the bottom, as well as the -2- near the left-center, each have only a ring of three hexes to work with. We can go ahead and solve them as we have been, by marking the endpoints and clearing the middle cells from each cluster.

So when we do this to the -2- near the left-center, we actually give a blue cell to the {2}. Because the blue hex is at an endpoint in its ring of four hexes, we know that the next one over from it has to be marked and the remaining two cells erased from the {2}. So we start with this:




You probably already see it; the "3" we uncovered a moment ago now has three blue hexes, which lets us clear two more from the puzzle. Notice now that the "3" we just revealed right above the -2- near the bottom left has only two remaining choices for blue hexes. Each of them will be shared by the -2-, eliminating the hex still attached to its bottom-right edge.

Let's move now to the top-right corner. See how the -3- has only a ring of four hexes around it? We know from past experience that the two endpoints must be marked. Go ahead and do this, which will also give a blue hex to that "1" on the right edge. Clear the other two orange hexes still bordering the "1".



We can actually solve that -3- now. Since the -2- we just revealed has a blue hex shared with the -3-, we need to eliminate the next orange hex over in the ring around the -2-. The hex we erase is also shared with the -3-. We open a "4" from this, and with only four active cells remaining around it, we can go ahead and mark all of them. Doing so also gives the required blue hexes to both the -3- and the -2-; clear the remaining orange hex from below the -2-.

Our next step isn't quite as obvious, but it isn't too difficult. The -3- just to the left has two blue hexes; notice that they are connected. So we need to erase the next consecutive cell in its cluster. When we do so, we also erase one of the hexes the "3" positioned directly below the -2- could have claimed for a blue hex, leaving only three active hexes around the "3". So now, we mark the other two.

We're still left with two possible choices to solve the -3-. However, notice that both of these hexes are shared with the "3" at the bottom-center. Since we know that one of these hexes has to be marked, the "3" cannot claim the orange hex on its top-left edge. We need to clear this cell next. This reveals another "4"; with only four active cells surrounding it, just mark its two remaining orange hexes. This will also give a complete set of blue hexes to the -3-, the "3" in the bottom-center, and the "3" just up and to the left. Clear the orange hexes still remaining around each of them.

We're almost done with the chapter now. Let's actually shift focus to the top for a minute. The "3" located two cells above the "4" we revealed a moment ago has only one remaining choice for a third blue hex, which will be shared with the "2" above it. This means the hex along the bottom-right edge of the "1" at the top will be erased and the very top-center cell marked to complete the section. Finishing the puzzle is now obvious; there are only three remaining orange hexes, and the "REMAINING" counter shows three blue hexes yet to color. Mark them all to complete Chapter 1.



Hexes Earned for Completing This Puzzle: 6

Chapter 2 reintroduces us to the idea of working with entire lines of cells. Remember that the numbers outside of a line or column introduce new conditions for marking the hexes within that line or column. However, the way the numbers are interpreted is exactly the same as if they were empty hexes. Here's a quick primer:

• A normal integer, such as "2": The line has this number of blue hexes within it; they may be connected or disjointed.
  
• A braced integer, such as {2}: The line contains that number of blue hexes, and they all must be connected, with no empty hexes between.
  
• A negative integer, such as -2-: The line contains that number of blue hexes, and they cannot all be connected.

The last condition bears a little more explanation. Let's say that there is a column of 12 hexes, with a -5- governing it. The -5- only tells us that all of the blue hexes cannot be connected; however, this does not mean that some of the blue hexes cannot be connected! You could actually have four blue hexes connected, a series of five empty hexes, and then the fifth blue hex within that line! Or, you could have two blue hexes, a series of seven empty hexes, and then the remaining three blue hexes. And so on. Let the other conditions on the grid, as well as with any other lines intersecting such a column, determine which hexes will be marked.




It can be tricky just to figure out where to start on this one. Most of the columns have a number restricting the hexes which can be marked. And we're given only two empty hexes: A "3" and a -3-. We can start with -3-; it has only the standard ring of four surrounding hexes, meaning the two endpoints will be marked. But notice that when we do this, we give a blue hex to the column governed by the {4}--at an endpoint, no less. We've just solved both the -3- and that whole column; since the next three hexes in the column have to be sequential, and because one of them will give the -3- a third hex, we'll get to eliminate two cells.

If you eliminated the cell directly below the -3- at the end of the last sequence, you'll immediately see that the "3" this reveals pops up in the center of four active hexes--with three already blue. Clear the fourth. You might also notice that there is a "2" at the top of that column, and that the column has only three cells. So with the first cell marked and the second one eliminated, just mark the last one.

Let's continue to the left, where we find a column of three cells headed with a "1". The first two cells are adjacent to the "3" next-door; so we know that one of those cells has to be marked if it will ever get a complete set of blue hexes. This means the bottom cell in the column has to be erased. The -2- that this reveals actually lets us solve the column and the "3". The cell directly above the -2- has to be erased in order to separate its next blue hex from the one below the "3". So now, we just mark the hex at the top of the column.

So let's try to solve the next column over, as well; this one is also governed by a "1". This one isn't quite as straightforward. We need to recognize that whichever cell is marked in this column has to solve both the "3" and -2- cells in the next column. We can safely eliminate the top hex of the column, then; it's obviously not going to complete them. So this gives us an empty "1" cell, with the hex on its bottom-right edge already blue. Go ahead and clear the other two orange hexes on its bottom rim. The "2" revealed directly below the "1" has only one choice to claim a second blue hex; mark it, then clear the bottom hex of this column.

Alternate Solution: If you understood the layout, you could have skipped pretty much that entire last paragraph. It's not hard to see that there is only one possible way to position the blue hex in this column so that it completes both the "3" and -2- cells. Marking it immediately allows you to clear all the other hexes in the column; revealing the empty hexes alone would then have led you to the same solution.

If you have any concept at all as to how Hexcells works, you don't even need my help in solving the first column with the {3} at the top. If you are still working through the mechanics, though, the only thing important to note is that when you have a gap between cells in the same row or column, it does not constitute a break in any chain of consecutive blue hexes. Only an empty hex is considered a break. That being said, we know that the "2" we revealed a moment ago has to claim the top two hexes of the column. Cross the gap and mark the next hex down to make three consecutive hexes, then erase the last one.

That solves the left side of the puzzle. What about the right? Well, we don't have anything quite as obvious as what we just worked with, but check out the "1" at the bottom of the column with the (4) at the top. Clearly, the cell above it has been marked, which lets us erase the orange hex along its top-right edge, which is in yet another column governed by a "1". Since we now have only one active hex within the column to deal with, we'll go ahead and mark it now. That gives a third blue hex to the "3" at the top, so now, just clear the two hexes on its right edge.

Here's a bit of a brain-teaser. The next column over from the cells we just erased is governed by a "1". There's no clear way of seeing which of the cells will be marked. But notice that we opened up an empty "1" cell a second ago; the only two active hexes bordering it are the top two hexes in the column we're working with. Since we therefore know that one of them will have to be marked, the bottom cell in the column can safely be erased.

The "2" that we just revealed gives us a critical clue and in fact lets us solve pretty much everything that's left except the final column. The "2" has only two active cells around it; when we mark both, we'll give shared blue hexes to all the other empty cells in the vicinity, giving them all complete sets. Clear the orange hexes that remain. The "1" we reveal at the top of the column headed by the "1" will pop up with its blue hex below it, so clear the next hex over, leaving only the final column to solve:

Only the final column, headed with a -2-, remains. We already know that the top cell has to be marked in order to give the "1" we just revealed a blue hex. So now, clear the hex below the cell we just marked. From here, just follow the chain of empty hexes to the solution.



Hexes Earned for Completing This Puzzle: 6

The last puzzle was a warm-up to working with lines of hexes. The next puzzle reintroduces the mechanic of line-marking. Remember that you can click on a line header to draw a white line through all of the hexes within that row or column. Clicking on the header a second time removes the line; right-clicking it dims the header if you're sure you are finished with it. (Don't worry; if you clear a line header prematurely, a second right-click will restore it.)




This puzzle not only reintroduces the line-marking mechanic, it throws a whole monkey wrench into the idea of working with lines to start with. I'm going to take a minute to isolate one of the lines to show you what I'm talking about; it's critical to understand this early.



Secondary (and, later, Tertiary) Line Headers can really complicate the conditions on the grid. We'll spend plenty of time on the mechanics and strategies for working with such divided lines of cells as we continue. Make sure that you study the previous image to at least gain a basic understanding of this concept before continuing.

To start the puzzle, then, it's a good idea to study the line headers to see if there are any obvious moves we can make. Every once in awhile, we'll encounter either a line headed by a "0", which is treated exactly like an empty hex with a "0", or a line whose number of blue hexes is equal to the total number of active hexes within. While we don't have a "0", studying the grid reveals that we do have one diagonal, headed by a "4", on the far right side of the puzzle with the same number of cells as the line header.

Returning to the grid itself, then, we have two opening moves. First, see how the -3- that I isolated in the last image has the standard ring of four blue hexes around it? We can go ahead and mark the endpoints in the chain, then mark all of the hexes in the afore-mentioned line on the right side of the puzzle.



You might notice that we marked a blue hex in the next-to-last column within the central grid. If you look up, that whole column is governed by a "1"; this means the other cells in the column can now be erased.

The -3- that comes up right above the blue hex is in an interesting position. It has five total active hexes around it, but two of the middle cells are already marked, and they are consecutive. Since its third blue hex cannot be connected to either of them, we now need to erase orange hexes on either side of the pair, then mark the only remaining active hex surrounding the -3-.

We can now solve two more columns. The hex that we just marked falls under another column governed by a "1". The top two hexes of the line can now be erased. Finally, the last column in this grid is headed by a "2" and contains only three cells. One is marked, and one was just erased; mark the top cell of the column to finish it.



Let's try to finish the central column of this grid now, which is governed by a "3". The top two cells in the column are going to be marked; they're the only active cells for the "2" at the top of the next column to claim. The second hex we mark will be shared with the "2" right below the one we just solved, eliminating the next cell down. This opens up another "2", which shares the same two blue hexes as the "2" on its upper-right edge. Clear the remaining three orange hexes around it, then mark the last cell in the column.

From this, we can now solve our first embedded lines of the game. Starting again from the right side of the puzzle, there's a diagonal governed by a "5". This line feeds into the central grid, where there is an embedded line governed by a "4". This means that four of the five hexes in the main line have to come from below the "4". Take a look; if you click the "4" to highlight the included cells, we can now see there are only four total active hexes within the embedded line. Mark those last two cells.

That last step also feeds into the line that I diagrammed out for you in the puzzle introduction. Click on both the Primary Line Header--the "5"--and the Secondary Line Header--the "2". Start from the "2" and work your way down; do you see two blue hexes? Erase the remaining two hexes within the embedded line. Now, remember that since we need five hexes within the entire length of this diagonal, we still need three more to come from above the "2". Notice that there are only three hexes in the upper line segment. Mark the others to complete this entire line.

One last thing. In the central grid, the column to the left of the one headed by a "3" is itself headed by a "1"; we gave it a blue cell a little bit ago. Go ahead and clear the top cell of the column to finish it.

The following image details that entire last series. There's a lot going on here; just take it slowly.



Special Note: I like to dim out the line headers for embedded lines as soon as they're complete, even if the Primary Line Header has not been fully solved yet. I also will tend to do the same thing with the Primary Line Headers. If I happen to solve the cells in between the Primary and Secondary line headers first, marking out the Primary Header helps me to focus on the remaining cells within the embedded line. These are simply personal preferences; experiement to see what playing strategy works best for you.

Well, if nothing else, we've solved the -3- in the first grid now; clear the lone remaining orange hex still touching it. This opens up a "4"; with only four active hexes surrounding it, we just need to color the fourth one. Just as importantly, though, it lets us solve another diagonal, and with that, two more columns.

Both the empty "4" cell and the hex we just marked fall into a diagonal governed itself by a "4". If we highlight the line, we can see that it now contains only four active hexes. Mark the two that have yet to be colored. Doing this actually gives the first and second columns of the central grid the remaining blue hexes they need. Clear their remaining orange cells to complete the grid.


We're almost done now. We can finally solve the first grid of the puzzle; if we highlight the diagonal in which those last two orange hexes fall, we can see that there are now only three active hexes within it. Since the diagonal is headed by a "3", this means we need to mark both.

Only the third grid remains. Go ahead and click on the "1" outside of that very first cell of the grid. Now look carefully; can you see that we've now erased every cell along that diagonal except for this one? Go ahead and mark it next.





The -2- cell in this grid will lead us to the solution. We gave it a blue hex fairly early on, at an endpoint within its ring of four hexes. Erase the next hex over from the blue one, since the second blue hex cannot be linked to it. That opens up a "3", but it more importantly solves the diagonal in which the "3" lies. This line is governed by a "2"; if we highlight it, we find that there are now only two active hexes within. When we mark the remaining hex in the line, we give the "3" the third blue hex it needs. Clear the two remaining hexes still bordering it, then mark the final hex of the puzzle to finish the level.



Hexes Earned for Completing This Puzzle: 8

See what I mean about how Hexcells Plus throws us into relatively complicated solutions early on? We're not even halfway through Chapter 2 yet and have already encountered a fairly intricate puzzle, with an introduction to lines embedded within larger lines. The game will get more challenging as we continue, and we'll be introduced to one more puzzle mechanic that may make you want to tear your hair out when we get to the final puzzles of the game.




I decided to play this one in a practice run before writing it out, and it's a fairly complicated solution. I had to sit and pore over it for awhile before I even figured out how to get this puzzle started!

The only obvious move we can make is to clear the four hexes from around the "0" on the right side. Let's start here. We get a couple of {2} cells out of this, each of which will have a continuous ring of three orange hexes around them. We know from experience that the middle cell in such a cluster will always be marked, so we can do that to get our first two blue hexes of the puzzle.

The blue hex we mark next to the upper {2} falls into a diagonal governed by a -2-, which means that the next blue hex in the line cannot be connected to this one. We can eliminate the cells on either side of it, then. This at least gives us a "2" in that sort of island of cells in the center.

Similarly, the blue hex we marked under the lower {2} falls into a column that is also governed by a -2- at the top. This at least lets us get rid of the hex directly below the blue cell. So we start with this:





So while that isn't much, it at least gets us going. Concerning that upper {2}, we've now eliminated one of the two orange hexes it could have claimed for a second blue hex; we can now just mark the last one to complete it, which also gives the "1" below a blue hex. The hex below the "1" can now be erased.

Note that when we erase that last hex, we eliminate one of the hexes that the lower {2} could have claimed. So once again, just mark the one that remains. Also, the "1" that we just revealed a second ago will share a blue hex with this {2}, so again, the hex below will be erased. That reveals a "2", which we can't immediately solve. The sequence, then, looks like this:


We've solved a reasonable amount of cells on the grid, so let's look at the line headers and see if they tell us anything. Look carefully; see the diagonal at the upper-left corner which is governed by a "2"? If we highlight the line, we can see that we have given it the two blue hexes it needs, which will eliminate two important cells from the grid.

In the top-left corner, we get a "0"; let's go ahead and follow it. We actually eliminate quite a few cells from the resulting sequence. Afterward, the "1" we uncover at the top of the final column pops up right next to a blue hex; clear the cell below it. We get another "0" to follow, eliminating a few more cells before we join back up with the column governed by the -2-.


So concerning this column: We have an empty "1" cell adjacent to both of the column's bottom cells, with no other active hexes around it. Obviously, one of them will have to be marked; we can now erase the top hex of the column. Doing so reveals a "2" with only two choices for blue hexes. Go ahead and mark them both, thereby also giving a second blue hex to the "2" on this one's bottom-left edge. Clear that cell's last remaining orange hex to reveal another "2" that will share the same blue hexes. This lets us erase the cell above it.

The "1" that we just revealed at the top of this column, which is governed by a "3", will share a blue hex with both of the empty "2" cells we just solved. Go ahead and clear the next hex to its left. Another "0" leads us to the central column, which is governed by a "5". Go ahead and highlight that column, then count the number of orange hexes. Did you count five? Go ahead and mark all of them.

We can solve a couple more lines now, as well. Going back to the diagonals marked on the left side, highlight the one governed by a "3". We can see that it now has only three active hexes; mark them all. Now, let's take the column headed by a "1" to the right of the central column. When we marked those last three hexes, we gave this column the lone blue hex it needed. Clear the rest from the column.

Let's try to solve our little island in the center now. The "2" at its top-center has just been given a second blue hex. Go ahead and clear the one on its bottom-left edge. Ah, this is easy; that "3" we just revealed has only three active hexes, so the last two in this group will be marked. As a final step for this sequence, then, notice that we just gave a lone blue hex to the column left of center, as well; this one is also headed by a "1", so go ahead and clear its final hex on the bottom.

Solving the island in the middle now lets us solve that -2- diagonal running from the right side of the grid. If we highlight it, we can see that it now has two blue hexes; clear the two orange hexes which remain. Now here's something you don't see often; that {2} we just revealed has three hexes, but one is on the opposite side from a linked pair! Guess which two will be marked?

The "1" positioned on the upper-right edge of the {2} is naturally going to share one of its blue hexes, eliminating the other two surrounding it. That puts us in position to solve the diagonal from the right side which is governed by an "8"; we now have eight active hexes, with only the last one needing to be marked. That solves the "1" in the bottom-left corner; just erase the hex right above it. Go ahead and follow the "0" that this reveals, too.

Let's try to finish this section, then. We have a trio of empty"1" cells; the one we just revealed, and the one next to it, have only one hex they can claim, which will actually be shared with all three of them. Clear the hex above the third "1" in the chain. That revealed a "2", but there are several choices for its second blue hex. However, we can solve it.

Go back to the top-right corner; we have a diagonal that itself is governed by a "2". Highlight it, and we find there are only two active cells in the line--and marking the second one gives that empty "2" cell the second blue hex it needs, clearing the other two from it. The "3" just revealed has only one choice for its third blue hex, and the last cell in the section will then be cleared. One last step here: The column headed by a "4" here will now have only four active cells; mark the last one on the bottom.

Only the bottom remains. The pair of empty "2" cells at the bottom-center shares both of those blue hexes, so the orange hexes immediately adjacent to them will be erased. Follow those two zeroes. The left side will now be easiest to solve first. The column in which the "1" we just revealed falls is governed by a "4"; it now has only four active hexes, so the remaining two need to be marked. Erase the hex on the top-left edge of the "1", since we just gave it a blue hex, and then the "3" this uncovers will claim the last hex in the section.

I anticipated this puzzle taking two pages; here's where we left off:



We're now left in a pickle. If we examine the remaining line headers and the pattern of empty hexes still left unsolved, there is no clear way to determine which cells will be marked. We're left with trying to eliminate at least one hex that will lead us to the final solution.

What's the simplest move? Well, we still have an empty "1" cell whose only two choices for blue hexes come from the column governed by a -2-. We also have two empty "2" cells, each of whose two choices for a second blue hex fall under a diagonal governed by a "4", with three blue hexes already marked.

There are two ways we can approach this. The absolute simplest move is probably to deal with the "1" and the column headed by the -2-. Let's run the possibilities. If we mark the top hex, that means the remaining two hexes in the diagonal governed by the "4" have to be eliminated; this would make it impossible for the "2" closest to the central ring to get a second blue hex. Problem solved; that cell gets eliminated. Mark the final cell in the column to complete it and solve the empty "1" cell.

We uncovered a "2" on the upper-left edge of the "1", but more importantly, the "2" above this one is left with only two active hexes. Marking the second will give our diagonal headed by the "4" its final blue hex. Clear the final hex from the line. That same blue hex also gives the "2" we just uncovered a second blue hex, and even the "1" we just revealed in solving our diagonal shares that same blue hex. Erasing the hex below this "1" gives us a "2" that shows us how to complete the puzzle.




Remember how I said there were two ways to approach this final section? Here's the second.

Let's say we started with that pair of empty "2" cells above the diagonal governed by the "4". We could examine each of them individually. For the first one: We know that its second blue hex would have to come from either the first or second orange hexes remaining in the diagonal. Since either would give the diagonal a fourth blue hex, by extension, the third orange hex in the chain would have to be eliminated.

Similarly, if we take the second empty "2" here, its blue hex would have to come from either the second or third orange hexes within the diagonal. Since either of these would also give the diagonal a fourth blue hex, by extension, the first orange hex in the chain would have to be eliminated.

So the center orange hex has to be marked; it has to be a shared blue hex to complete each of the empty "2" cells.



This completes the most difficult puzzle of Hexcells Plus so far. And we are just now at the halfway point of Chapter 2...

Hexes Earned for Completing This Puzzle: 8

We're halfway through Chapter 2 and have already encountered quite the difficulty spike! Chapter 3 unlocks at this point if you would like to take a look at the upcoming challenges.


We get only one solid cluster of cells in the center of this puzzle; this rest is all isolated, individual cells:



This is one of those puzzles that can make you think, "OMG, there's no way I can solve this!" But truthfully, once you get a few moves in, it's not that bad.

The hardest part is probably just getting this puzzle started. There isn't a whole lot of togetherness here; we get only a single island of continuous cells in the center, with everything else just being individual, separate hexes scattered around. Check the line headers carefully; you should see a diagonal marked on the right side by a "5". Go ahead and highlight the line; you should find that there are only five active hexes within it. So now, go ahead and mark the entire line.

We just placed a blue hex into the central column, which is governed by a -2-. We know that the column header ignores the gaps in this line, meaning that the next cell up, at the bottom of our island, has to be eliminated. This reveals a "0", so we can at least erase the next cell up the column. We can't, however, solve the "2" that this uncovers yet.

The next major move we can make has to do with the diagonal headed by a {6} on the right edge, which basically cuts the puzzle in half diagonally. There are nine total hexes in the line. We don't have endpoints yet, but if we were to number and count out the possible combinations, we would find a few guaranteed hexes. You can write down the combinations if you wish; I'll diagram them here:

1, 2, 3, 4, 5, 6
2, 3, 4, 5, 6, 7
3, 4, 5, 6, 7, 8
4, 5, 6, 7, 8, 9

So we basically find that the middle three hexes in the line--Cells 4, 5, and 6 if we're numbering them--are guaranteed to be marked, regardless of where we establish the endpoints. Marking those three cells, then, gives us this to open the puzzle:





If you're paying attention, you probably already know where we're going next. We've just given the central column the second disjointed blue hex that it needs, so we're now going to erase its remaining orange hexes. The "3" that we come up with at the top of our island will claim the only remaining hex it can; similarly, the "2" at the bottom of the island will lose its remaining orange hex since we gave it two blue cells with the last sequence.

So we just revealed a "4" here, and it is surrounded by only four active hexes; go ahead and mark the other two. Let's now look around the puzzle and see if we can solve any more lines. Check the diagonal governed with a "1" on the right side, which is right below the one with the {6}. If we highlight it, we can see that it has now been given the lone blue hex it needs, which erases the rest of the line.

One last step for this sequence. Just right of center is a diagonal governed by a {2}. We just stuck an empty hex between that lone cell and the rest of the line. So go ahead and erase this one, too.





Let's continue trying to work the lines now. Start with the very top-center hex; it marks the endpoint for two diagonals. Extending to the left is one governed by a {3}; extending to the right is one governed by a "2". Go ahead and highlight both, but start with the one governed by the "2". We can see it now has only two active cells, with one already marked; marking the second solves the whole line. Notice that this gives us another blue hex in our chain of six required for the central diagonal; we'll come back to this shortly.

Now, let's take the top cell's other diagonal. It needs a chain of three continuous blue hexes, but there are still four active cells in the line. That's okay; we know the middle two must be marked. Go ahead and do this now; we'll figure out the endpoint soon.

If you marked the cells from right to left like I do, look at the cell directly above the one you just marked; another diagonal, this time governed by a "5". If we highlight this line, we'll see that we've now given the line a fifth blue hex. Erase those that remain.




Let's try to solve our little island now. In its top-left corner, that "1" has a blue hex, clearing the cell below it. The "3" revealed claims the final hex on this side. Well, well, well; if we look carefully, we can see that we have our endpoints for this diagonal's chain of six blue hexes! We left a gap in it, which fittingly corresponds to the final cell on the island. So now, mark the island's final hex, then eliminate the remaining orange hexes in this diagonal.

The puzzle is basically solved now, but we want to take the final lines methodically. I'll do one more break before the final solution to make sure everything is clear. Let's move to the diagonal governed by a "3" at the left-center of the puzzle. Highlighting the line shows that we have only three active hexes now. Just mark the ones which remain.

The next diagonal up is also governed by a "3"; this one has the three blue hexes it needs, so just erase the final orange hex within.







We're done at this point, and you can probably already see the solution. Returning to the cell at the top-center of the puzzle, we gave the diagonal headed by a {3} the third blue hex it needs. Erasing the last one in the line leaves just two remaining orange cells. The "REMAINING" counter shows two blue cells left to mark; color these last two hexes to clear the level.



Hexes Earned for Completing This Puzzle: 9

We're nearing the end of Chapter 2 in Hexcells Plus! The next puzzle starts us out similarly to the last one.




While this puzzle does give us a large, connected grid of cells, we are again only given a bunch of line headers to start us. Our choices for opening moves this time, however, are a little more straightforward.

As always, we want to try to find the simplest moves to start out. Examining the column headers, we actually have two columns headed by zeroes. That's going to clear a slew of cells right away. Now, just to the left of the first "empty column" is a diagonal governed by a "6". Highlighting it shows that it contains only six active cells, so just mark all of them.

The final cell we marked just now, at the bottom-right of the grid, falls into a column governed by a {3}. Well, we just marked the bottom cell of the column; since we have an endpoint, just mark the next two up the line and erase the column's remaining hex.

One last thing: The central column is governed by another {3}. It has only five total hexes, so the center cell has to be marked. It's pretty easy to see that it will be included in all three possible combinations for three continuous blue hexes. We'll do some cleanup work here in a minute, but our productive opening looks like this:



This gets the puzzle off to a great start. We want to now do some cleanup work, and make some obvious moves that opened up to us during that last sequence. The next-to-last column, for example, is governed by a "1"; when we solved that diagonal a moment ago, we gave this column the only blue hex it requires, so we can clear the rest of the hexes within it. Go ahead and follow the "0" that we'll reveal at the top. The "2" revealed from clearing the next cell over will have only two possible hexes to claim; go ahead and mark them now. Now, at the bottom of the column we just solved, we revealed a "2" which already owns two blue hexes; clear the remaining hex to its left. The "1" this reveals will have only one possible hex to claim.

We now want to tackle some of the empty "2" cells we revealed when we solved the empty columns at the beginning. Notice how several of them have only two hexes to work with. Go ahead and mark all of them now. As a final step for this sequence, let's also solve the "3" that was revealed in the second empty column just down from the top. It has only three orange hexes to work with, so mark them next.

It may appear that we've hit a logical dead end here, as there's nothing truly obvious from either the empty hexes on the grid or the line headers as to what we should mark nex. What we need to do next is actually eliminate a cell around that -2- in the second empty column. We've given it a blue hex on its upper-right edge, so the cell below the blue one needs to be erased. That reveals a "2" with only a single choice for its second blue hex.

Let's move now to the column to the left of the -2-, which is governed by a "4". The column already has three blue hexes; two of its remaining three orange hexes border the -2-. Since we know one of them has to be marked, the top cell of the column can safely be erased. Doing so gives us a second -2-, only this one has a standard ring of three active hexes. Since one of the endpoints in the ring is marked, erase the middle hex and mark the third. Now, concerning the "1" on the upper-right edge of this -2-, go ahead and mark the remaining hex it can claim. And for the "3" that we just revealed on its bottom-left edge? It, too, has only an obvious choice for claiming its final blue hex.

A couple more moves opened up with that last sequence. Notice that to the right of the center column is a diagonal which is also governed by a {3}. We now have an endpoint for its chain of three blue hexes; just mark the next two in sequence, then clear the line's final orange hex.

Now, within the central column, we have two of its required three consecutive blue hexes. We don't know which is the endpoint yet, but we can now see that the bottom hex in the column falls outside of what is possible. Let's erase that hex next. Another "2", and again, the choices for its blue hexes are obvious.

But look what marking those hexes did. The column to the left is also governed by a {3}, and we just marked the bottom cell of that column. Now, mark the next two up the column, then clear the rest of its cells. When we get to the top, we reveal a "2"; and yet again, its final blue hex comes from the only pick available to it. But now, we just gave the central column its third consecutive blue hex! The final cell in this column can now be eliminated.

We revealed a "3" near the bottom of the central column; let's go ahead and give it the third hex it needs (there's again only one choice). The column in which that blue hex falls is governed by a "4", and we just gave that column the final blue hex it needs. Let's erase the remaining orange cell within. Another "3", again with only a single choice for its third blue hex. When we mark it, however, we'll also give that -2- from earlier the second blue hex it needs. Now, we can erase its remaining orange hex. Consequently, this also solves that whole column, which is also governed by a "4".

Three more diagonals can now be completed. The one farthest to the left is governed by a "3" and now has three blue hexes, allowing for the elimination of two more cells. The "2" we reveal by clearing the line's first hex is again in a position to claim the only two active cells bordering it. Consequently, this will solve the next diagonal over, which is governed by a "4", by giving it the final blue hex it needs. Finally, the next diagonal over to the right, which is headed by a "5", now has only five active hexes, with only one remaining to be marked.

Our next steps are going to take some planning. Let's make an easy move to start: See that "1" a couple of spaces in from the top-left corner? It has only one neighboring orange hex to claim, so let's mark it now.

We now want to take a look at that empty "3" cell that we have done absolutely nothing with so far. We have a bit of a dilemma with this one. Two of its orange hexes fall into a column that is governed by a "5", which already has four blue hexes. We know that only one of them can be marked; this means that the two hexes on its left edge will definitely be marked. That's the easy part.

The hex that we just marked on the upper-left edge of the "3" falls into a diagonal governed by a "4"; we just gave the row a third blue hex. Its two remaining orange hexes each border an empty "2" cell in the first column. Its only other active hex is above it--and it's in a column governed by a -2-.

Well, since either of the orange hexes on the empty "2" cell's bottom edge will complete the diagonal, we know that only one of them can be marked. This means that the cell above the "2" has to be marked. Now, given that this is a -2- column, the top-most cell of the column cannot be marked. When we erase it, we reveal a "3" with only three total active cells around it. So now we have to:

1. Mark the bottom hex of the first column.
2. Mark the top two hexes of the second column to complete the empty "3" cell.
3. Clear the bottom two hexes of this column, since we just gave it the two blue hexes it needs.

We're almost at the end of another fairly involved solution; here's where we left off:



So at the end of the last sequence, we both revealed a "1" at the bottom of the second column, and gave it a blue hex. The cell on its bottom-right edge can now be eliminated. We reveal another "1" from this, but more importantly, we can solve the diagonal in which it falls. On the right side of the grid is a diagonal governed by a "5". When we highlight it, we see that it has only five active hexes. Mark the final two orange hexes in the line.

The puzzle is now solved. We just gave the empty "3" cell we've been working with the third blue hex it needs. Erasing its final orange hex leaves us with only two orange hexes on the grid; the "REMAINING" counter shows that two blue hexes still need to be marked. We know what this means; mark the final two hexes to complete the puzzle.



Hexes Earned for Completing This Puzzle: 10


Well...At least we get some empty hexes to help us this time.



This is quite a difficult puzzle, especially for it to still be relatively early in the game. If you've already tried (and failed) a few times to perfect this one, you're in good company. I'll try to present the solution in the most straightforward way I can.

Let's go ahead and deal with the hexes around the "0" on the right side. We can go ahead and solve the "2" that appears on its bottom-left edge as it has only two orange hexes to work with. Go ahead and mark both.

Two more observations will open up a chunk of the puzzle. Let's look at the top section. We get two regular empty "2" cells, and two -2- cells. The two -2- cells have the same general pattern: Two adjacent orange hexes on one side, and a single orange hex on the other, which happens to link them together. We know that their blue hexes cannot be linked together, which means only one hex in each orange pair can be marked. Consequently, that means the cell linking the -2- cells together has to be marked. So we use this to mark the top-center hex of the puzzle, which also falls into a column governed by a "5".

Now, let's take the two columns in which the -2- cells' remaining orange hexes fall. Each column is governed by a "1". Since we already know that one cell of each pair has to be marked, we also know that the cells we mark will complete each of their respective columns. This allows us to eliminate everything within each column below these two pairs of orange hexes. We now start with this:

We can mark four hexes along the bottom now. Each pair of empty "2" cells that we opened up from our column-cleaning just now has only two orange hexes apiece to work with. We can safely mark all four of those cells.

We can also solve one of the diagonals. Over on the bottom-right of the central grid is a diagonal governed by a "3", into which our "0" falls; if we highlight it, we see that it now has only three active hexes, with two already marked. Let's finish it by marking the third one.

Staying in the central grid now, we revealed a -2- just to the left of the two blue hexes we marked in the beginning. Since we have a blue hex on its bottom-right edge, clear the one on its top-right edge. This reveals a "3" with only three active hexes bordering it; mark the two that it still needs. In doing so, we'll give shared hexes to the nearby "2" and to the "1" on the right edge of the grid. So let's next erase the hex above the "1". That reveals a "2" with only one obvious choice for a second blue hex. We still have one orange cell in this group. Notice, however, that it falls into a column governed by a -5-. If we highlight the line, we can see that there are only five active hexes within it. Just mark its last orange hex to solve it.

The next-to-last column is governed by a -3-. This actually gives us an important clue; since the column has two consecutive blue hexes within the central grid, we need to erase the next hex up the column. This opens a "1". It may seem like we can't solve it yet, but we can. Notice how the "2" on its top-left edge shares the same pair of orange hexes as the -2- over to the left. We've already established that only one of those hexes can be marked. This means that the hex on top of the "1" has to be marked, which will eliminate the cluster's remaining hex on the right edge. And now, since we've given the -3- column its remaining blue hex, its two remaining hexes at the bottom can now be erased. Once we do this, we can go ahead and mark the final hex in the last column to complete the pair of empty "2" cells we just revealed.

We need to turn attention now to the column governed by a "4" which lies directly to the right of the central column. Highlight the line; it has six total active hexes, with one marked blue. So we need three more blue hexes. To determine our next step, we want to use the empty hexes in the central grid to either find an obvious cell to mark or to eliminate. The top two hexes border a "3" which already owns two blue hexes. So we know that one, but not both, of these cells has to be marked. We get the exact same situation at the bottom, as well. The middle cell of this chain, if marked, would complete both the adjacent "2" and -2-.

We can use the logic governing the two empty "3" cells to develop a different kind of counting scheme. This is something that we will use later, as well; when the time comes, this concept will be explored in much more detail. But suppose we start counting multiple orange hexes as a single unit, as opposed to just counting each individual hex as a single unit. The target is to come up with a total of four blue hexes within this column. We've already determined that either of the top two hexes, and either of the bottom two hexes, is 100% guaranteed to be marked. Once we've determined which cell in each pair needs to be marked, its remaining hex will be eliminated. We already have one blue hex; let's count the two pairs of cells as 2 and 3. Where does the fourth hex come from? From the center hex of the chain, of course!



What this establishes is that the middle orange hex in this line has to be marked, also completing the "2" and -2- along its right edge. This erases the cells above and below the one we just marked, leaving only the top and bottom hexes in this line to be marked to complete the whole column.




Alternate Solution: If you want, you can also just run through the possibilities of which cells to mark. You only need to go through a couple. If you took the top cell in the line, for example, you would easily be able to make a case for it to be marked by simply going through the remaining relationships, which would tell you what to mark and what to eliminate. If you took the next cell down, however, you would run into problems if it were to be marked. Marking it would erase the cells above and below it. The next cell down, then, would have to be marked to complete the -2-, but it would also complete the "3" at the bottom of the next column. The column's final orange cell would have to be erased, and it would be impossible to give the column four blue hexes. This logic would allow you to both mark the top cell, and to eliminate the one below it. You would then be able to arrive at the same solution using the relationships between the hexes on the grid.

After a fairly intense first session with this puzzle, here's what we have so far:



We can complete another diagonal now. Over on the right side of the central grid is a second line headed by a "3". Highlight it, and we can see that there are now three blue hexes within it. Go ahead and erase that final cell. Would you believe that this reveals a -2-? And with a standard ring of three active hexes, too. One of its endpoints is already marked; mark the other one and then erase the final hex in the cluster.

Solving this diagonal also helps us to solve the one right above it. You certainly saw the line governed by a "4" a moment ago; we just solved enough cells in this line to reduce it to four active hexes. Two of them are already marked, so just mark the two which remain.

Let's now try to solve the central column, governed by a "5". We've given it three blue hexes now. When we completed that last diagonal, we gave a third blue hex to the "3" in the next column to the right, near the bottom of the central grid. We have one orange hex to clear from this "3". Interesting; another "3". Go two columns to the left; see that -2-, as well as the blue hex on its bottom-right edge? The cell above this blue one is the next hex we need to clear.

So when we erase that cell, we get yet another "3", only this time, there's only a single remaining choice for its third blue hex. When we mark it, we'll also give the "3" we were just working with the third blue hex it needs. When we clear the hex right above that "3", we reveal another "3", but more importantly, it solves enough cells to reduce the whole center column to only five active hexes. So now, color the remaining two.

Doing this also gives the "3" we just revealed the final blue hex it needs. Erase the orange hex on its top-left edge. Would you believe that we get another "3"? This one, however, is also in a place that gives it only one possible hex to claim for a complete set of three blue hexes.




We've solved about 80% of it now. This is a good time to recheck the remaining line headers to see if any others can be solved (always do this after solving a decent number of cells). Starting with the diagonal headed by a "4" at the very upper-right edge of the board, highlighting it shows that we've reduced it to only four active cells. Now, we just need to mark the two which remain.

When we do this, we also give the "1" at the left edge of the central grid the blue hex it needs, and we now can clear the two orange cells still bordering it. We may not be able to solve the empty cells this reveals yet, but we can solve two nearby diagonals. The line governed by a "3", into which the "1" we just solved falls, has only three active hexes, with one still needing to be marked. Additionally, the diagonal headed by a "3" directly above this one already has three blue hexes, letting us clear the lone orange hex still remaining within it.

One more diagonal to work for right now: At the very upper-left corner of the puzzle is a line headed by a "2"; the line already has two blue hexes, so those two orange cells in the top grid can be cleared. This should help lead us to the final solution of the puzzle. Clearing these two cells leaves only one hex for the two empty "1" cells to claim. The cell that we mark becomes a focal point; it solves both the column and the diagonal in which this cell resides. Since both the column, governed by a "2", and the diagonal, governed by a "4", have their required blue hexes, the remaining cells within each line are now erased.

We should be able to finish the puzzle now. Let's go to the top-right corner now, to another diagonal governed by a "2". We'll now find that it has two blue hexes, as well; erase the line's two remaining orange hexes.

We could use the remaining line headers to solve the puzzle now, but we don't need to. There are now only four orange hexes left on the grid; the "REMAINING" counter shows four blue hexes yet to mark. So mark all four remaining hexes to leave Chapter 2 of Hexcells Plus in the dust!



Hexes Earned for Completing This Puzzle: 10

Wow. We are only just now finished with Chapter 2 and have already had some fairly complicated solutions to work through. Chapter 3 will really test your ability to examine line and cell relationships. We'll also revisit embedded lines in this chapter. All of this will prepare us for the final puzzle mechanic that Hexcells has for us: Grid overlays.




It's a giant cheese ball!!!

Hopefully that made you laugh a little bit. This puzzle is actually one giant hex made up of many smaller hexes. We do get a fairly large number of empty hexes. All of them should tell us at least how many blue hexes they need. Right?

Um, wait, what are those three empty hexes in the center? Question marks? What do you mean "Question Marks"?????

This is a dirty trick introduced to us in Hexcells Plus. For the rest of the game, we will see empty hexes with nothing but a "?" inside. There is absolutely no way to reliably determine how many blue cells such a hex will capture. You must use the other conditions on the grid and any associated line headers to fill in the blanks.

Luckily, this puzzle gives us a fairly gentle introduction to this concept by confining the "?" cells to the center of the puzzle, with only a few orange hexes around them. We'll work our way back to them.

Looking around the grid, there isn't much we can do with the line headers yet. However, three of the -3- cells are surrounded by a standard ring of four orange hexes. We know for each of them that the two endpoints have to be marked. So let's start by doing this to at least get a few guaranteed blue hexes to begin.

One of those -3- cells is near the top-left corner of the grid, and one of the blue cells we just defined around it falls into a column governed by a -2-. Go ahead and clear the hexes above and below it. Doing so eliminates one of the hexes from the -3-, and we can now fill in its last remaining hex. Now, that last blue hex gave the "2" we just revealed a second blue hex, letting us eliminate the three remaining orange hexes around it.

One last thing we can do with this column headed by the -2-. Notice that at the bottom of it, we have an empty "3" cell with four active hexes around it, two of which fall into this column. We can't give the cell three blue hexes without one of these being marked. So we can now mark the two hexes on the right edge of the "3", then clear everything from the column above its remaining orange hexes.

Let's look now at the "4" we revealed a moment ago, just about halfway down this column. It has only four active hexes--two on the left, two on the right. Mark all four. Notice that we just gave the "3" right above it a set of three blue hexes. Clear the one on its top-right edge; the "2" it reveals pops up between two blue hexes, letting us erase the two orange hexes on its right edge. The -2- right above that "2" now has only one choice for its second blue hex.

There are a few things we can do from here. First, we can give the "2" at the top of the -2- column a second blue hex, also shared with the "2" on its bottom-right edge. Go ahead and clear the remaining orange hex from the second "2". We can't solve the "4" that this reveals just yet.

Let's move now to the "2" beneath the -3- in the third column. It has two blue hexes, erasing the one on its bottom-left edge. The {2} that this reveals is very helpful. The blue cell on its bottom-right edge is at an endpoint; mark the next one over, then clear its remaining two orange hexes. The very top-left cell will then be marked to solve its adjacent empty hexes. Finally, for the "2" we revealed on the bottom-left edge of the {2}, mark the cell directly below it; again, it's the only choice remaining for its second blue hex.

We've done about all we can with this side of the grid. Let's move back towards the center, to the "1" positioned on the top-left edge of the "?" in the central column. Since it already has a blue hex, let's clear the hex directly above the "?". This clears a hex from the "2" positioned above the "1", leaving only one choice for its second blue hex; go ahead and mark it next.

We'll need to work back to this area later; there are no more logical moves we can make in this quadrant. Our opening revealed some steps we can take on the far right side of the grid. That "1" at the bottom-right, for example, has its lone blue hex. We can go ahead and erase the hex on its bottom-left edge, which also clears one from the -3- directly above. Now, the -3- has only a standard ring of four active hexes; let's mark the second endpoint.

We just gave the "1" we revealed a second ago the blue hex it needs. Erase the orange hex on its bottom-left edge. The "2" we reveal cannot be solved yet, however.






Move up the final column now, to the {2}. As with the one before, we have a blue hex at an endpoint in its surrounding ring of active hexes. Mark the next cell in sequence and clear the other two.

Interestingly, we reveal another {2} on the upper-left edge of the first. We have the exact same situation, too: A blue cell at an endpoint of its surrounding ring of four continuous active hexes. So we'll solve it the same way, too.

This time, we reveal a -2- on the upper-left edge, and another {2} directly above this one. The {2} has only three active hexes around its top rim; we know from this that the cell directly above it has to be marked.

For the -2-, however, we want to clear the next hex over from the blue one. Be careful here! Wow, another -2-. Its positioning leaves it five active hexes. Again, the blue one marks an endpoint. Clear the next hex over in this chain, as well. We can't do anything with the -3- that this reveals yet.




No doubt that you've already seen our next move, and if you've already made it, good!! That puts you a little bit ahead! This walkthrough is simply taking a more methodical approach due to the puzzle's complexity. Anyway, we want to go ahead and mark the top-right hex of the grid now; it's the only one that the "1" below it can claim, and it solves the {2} we revealed a couple of steps back. The hex on its top-left edge can now be erased.

This is a very important elimination; the -2- on the bottom-left edge of the {2} we just solved now has only one remaining choice for its second blue hex. That hex will be shared with the "2" right above the -2-, clearing two more hexes. Each "1" revealed from these already has a blue hex apiece, erasing three more cells from around them.



We'll make one more selection along the top edge before moving back down. The "1" at the very top of the column governed by a "4" has only one choice for its blue hex, so mark that one next.

Let's actually move down the column that this "1" resides in; the -2- four cells down has the two blue hexes it needs, so we can clear its two remaining orange hexes. The "2" which pops up on the bottom-left edge of the -2- has only two choices for its blue hexes. Once these are marked, we will actually give a shared blue hex to both the "2" above this one, and to the "2" located above the "?" in the middle. This lets us clear the next hex up the column governed by a "5" (it's the fourth cell from the top in this column).

We're a little over halfway through with this puzzle now. Here's where we left off:



We'll return to finish what we just started on top here shortly. We have done virtually nothing with them so far, but we can now solve three full lines.

Let's start with the column governed by a "3" in the same area we've been working. Highlight it, and we now find the column has the three blue cells it needs. Erase the last four near the bottom of the column.

The criss-crossing diagonals governed by a "4" are in the same situation. They start near the upper-left and upper-right corners of the grid. Each have the four blue hexes they need, allowing us to clear their remaining orange hexes, as well.





Let's go ahead and try to finish the top now. We revealed a "3" a few moments ago in the column governed by a "5". Let's give it the remaining blue hex it needs. That blue hex is actually shared with both empty "2" cells in the next column. Go ahead and erase the cell above the one we just marked.

The dueling empty "4" cells here are in an interesting position. For the one on the left, two of its three orange hexes are along a diagonal governed by a "1". Since we know only one of these can be marked, the cell on its bottom-right edge will have to be marked. It will also be shared by the other "4".

We've actually reached a logical dead end; there's no way to narrow down how to complete these cells without guessing, so we'll try to fill in the bottom and come back. The bottom-left quadrant of the puzzle has a little less to deal with than the bottom-right quadrant. Start with the third column, at the "3" located four cells up from the bottom. It now has only three active hexes, with two still needing to be marked. Do this next.

Both hexes we just marked are shared with the "2" on the bottom-left edge of the "3". We can clear the two orange hexes still bordering it, leaving only a single choice for the "1" on the left edge to claim.






Let's continue in this general area now. There's another "3" in the third column, two cells up from the bottom. It only has three active cells, with two needing to be marked. Marking them does several things. It completes all of the empty cells in the area, and it also, finally, gives the -2- column a second blue hex. The final orange cell in the column can finally be erased.

Clear the orange hex to the right of the "1" at the bottom of the column, then go halfway up the column to the "3" that is positioned right above the -3-. As with the last few, it has only three active hexes around it. Marking the one it still needs also gives a shared blue hex to the -3-. We can now erase its final orange hex, as well.



Continuing along the bottom edge, then, the "1" we just revealed naturally shares the blue hex above it with the "1" we just solved. Erase the two orange cells to its right. Give the "2" that we just uncovered the only remaining hex it can claim to also give that -3- a few spaces over a third blue hex. Erase the orange hex still on top of the -3-.

We still have some missing information for filling out the bottom. The next move we can make to that end takes us to the "3" positioned on the bottom-right edge of the "?" in the column governed by a "5" to the right of the center. That "3" has only three active hexes, with two still needing to be marked. From that "3", move directly to the right two spaces to the "4"; it has only four active hexes, with three still needing to be marked.

Yikes. We've filled in almost everything and still don't have a clear direction of where to go to complete the puzzle. Let's try to find the simplest move. The column governed by a "4" on the right side has only five active hexes left, with only two actually marked. This might be our best bet.

The bottom two hexes of the column both border the "2" at the bottom of the next column. The "2" already has a blue hex, so we know that only one of these cells can be marked. This actually guarantees that the orange hex above this pair will be marked.

Marking this hex actually solves the "3" on its top-left edge. That "3" still has an orange hex on its bottom-left edge that we can now clear, revealing another "2".






Believe it or not, that elimination actually gives us some insight into how that "2" in the central column will be solved. The cell on its top-left edge cannot possibly be marked. Here's why:

First: The orange hexes on its bottom and bottom-left edges are shared with the "3" at the bottom of the left-most column governed by a "4". The "3" has two blue hexes; whichever cell completes the "3" will also be shared with the "2". This makes one blue cell for the "2".

Second: The orange hexes on the top-center and bottom-right edges of the "2" are shared with the "2" we just revealed. That "2" already has a blue hex; its second will be shared with the "2" we're trying to solve, making its second, as well.

Thus, the hex on the top-left edge of the "2" will now be erased, which also clears a hex from the nearby "4". That leaves only one remaining hex for the "4" to claim for a complete set.






We can now solve the column governed by a "5", which should open up the rest of the solution. The column now contains only five active hexes; mark the two which remain.

We'll try to work from top to bottom now. We just gave the "4" on the right side at the top a fourth blue hex. Erasing its remaining orange hex reveals another "4", only this one has only four active hexes around it. The top two cells will now be marked to complete that section.

We also can now solve the column to the right that is governed by a "4". Of the two remaining orange hexes here, the top one is now eliminated; the "2" on its top-left edge has the second hex it needs. That leaves only the bottom hex of the column to be marked.




We're now down to the final blue hex. Highlight the column headed by a "4" to the left of the center; we've left it only four active hexes, with one left to mark. That's our final blue hex. Mark it, then erase the last three hexes to complete this monster of a puzzle.



This puzzle can be brutal, even if you know what you're doing. Getting it started isn't so much the issue as it is paying attention to all the intricacies. It establishes more difficult relationships between the cells on the grid, and even the line headers don't come into play as much until later on. This doesn't diminish the importance of referring to the line headers in this or any other puzzle; this is just a case where they didn't play much of a role until late into the solution. This puzzle is a bit of a marathon, and there are many places where it seems like there isn't a logical move to make. Truth be told, this solution may be clunkier than the most straightforward way of approaching it; try other approaches and see if you find something that works better. This solution will work. It's actually important to understand the intricacies involved as the later puzzles become still more complicated than even this one.

Hexes Earned for Completing This Puzzle: 14

Welcome back to "Brain Drain," or "How to Go from Zero to Insanity in 60 Hexes or Less"!




Pun aside, this is actually one of the more interesting puzzles from a design perspective. We get seven bubbles of hexes, with numerous columns and rows defined by line markers. We're even given an entire column for free! I can't think of another puzzle in all of Hexcells that does this; it's almost as if it's daring us to solve it even with an entire column solved for us...

And yet, solve it we shall! Let's just start with the middle bubble; we can already see that the "2" in the very center of the board has the two blue hexes it needs, letting us erase four orange hexes right off the bat. Of the empty cells we revealed, take the {2} first. As we have seen, it is positioned in the middle of a ring of four consecutive active hexes, with one of the endpoints already marked. Mark the next in sequence and then clear the others.

The "3" we revealed right above the {2} is left with only three active hexes along its top edge, so go ahead and fill in the two which remain. Now, move to the -3- on the bottom-right edge of the central "2"; we know that any -3- with only four continuous active hexes will have the endpoints in the ring marked. Mark the second one, thereby also giving the "2" that we just revealed a second blue hex. Clear the two orange cells at the upper-right of the bubble. As for the third cell that the -3- needs? Well, we can't solve that yet...

So in a practice run, I decided to try to solve the top bubble next. Stay with me on this one because it's almost completely solvable, as well.

It's easy to get thrown off on this one. We have a "4", with five active hexes surrounding it, located right above a "1", which has four active hexes surrounding it. Let's look at this carefully. We're given a free blue hex above the "4", leaving four orange hexes around it. Two of these are shared with the "1", and that's the trick to this section.

Obviously, the "4" has six total hexes around it; the sixth, of course, is the empty "1" cell. Well, it's impossible to give the "4" a complete set of blue hexes without also sharing a blue hex with the "1". Which means that of the four orange hexes surrounding the "1", there's no possible way for it to claim the hexes on its bottom-left or bottom-right edges; those are the cells to eliminate first.

The "4" that we just opened up on the bottom-right edge of the "1" is in the perfect position; it has only four orange hexes to work with, so we're going to mark all of them. And that move alone gives the "1" the blue hex it needs; when we clear its final orange hex, we can solve the "4" at the top. Additionally, all of the bubble's empty "3" cells have only three total active hexes apiece to work with. Once they're all marked, only one hex in this bubble will be left unsolved.

Let's turn attention to the bottom bubble next. Both the "3" on top and the "2" on the bottom have only obvious hexes to claim; mark them all next. We can now turn our attention to the line headers for the first time. Off of the bubble on the bottom-right edge of the board is a diagonal governed by a "1". When we highlight it, we can see that the blue hex we marked on the right side of the "2" feeds into this line, erasing all of the line's remaining orange hexes.

Now, the "3" that we just revealed on the bottom-right edge of the bubble has only three active hexes, with one already marked. So now, just mark the other two. This gives the -3- in the center of the bubble a second blue hex in a rather interesting position. We know that the cell on its upper-right edge cannot be marked, because it would link all of its blue hexes together. So erase this one next. The "4" uncovered already has four blue hexes; clear the orange hex on its upper-right edge.

We need another diagonal to make our next move in this bubble. Again moving to the bottom-right bubble, we've got a diagonal governed by a -2-. This one runs through a hex we marked for the "3" a few moments ago. The line has a consecutive orange hex at the end which must now be eliminated. Well, the "2" that this reveals doesn't help all that much. We may need to come back here in a little bit.

We opened up a few things in the bottom-right bubble; let's see if we can do anything here. Interestingly, the "2" which popped up between the pair of empty "1" cells when we erased that diagonal has only two hexes it can claim; naturally, each will go to one of the empty "1" cells. When we clear their remaining orange hexes, we incidentally solve the diagonal governed by a "5" from the edge of this bubble.

Our next move actually starts from the bottom-left bubble, along the diagonal governed by a "3" along its top-left edge. Highlighting it reveals that we have already given the line three blue hexes. Erasing its remaining orange hexes reveals a "0" that lets us erase several more hexes.




So in the bottom-left bubble, the "2" which appeared in its next-to-last column has only two hexes it can claim, which will be shared with the "3", the "1", and the second "2" in the same region. The remaining hex below the second "2" can now be cleared. This reveals a "1", which also shares one of the blue hexes; clear the orange hex beside of it, too.

Both the "3" we just revealed, and the one in the center of the bubble, have only obvious blue hexes to claim. After marking them all, move to the diagonal governed by a -2- in this bubble; we've just given the line the second blue hex it needs. Now, erase the line's remaining two orange hexes. This gives us a "4" in the bottom bubble, with only four total active hexes around it. Mark the two which remain.

Now, to complete the bottom bubble, we need to go all the way to the top. The first column in the top bubble is governed by a "5", and it extends all the way to the bottom. The line has five blue hexes already; the final hex in the bottom bubble now gets erased.

In the bottom-left bubble where we just were, the diagonal governed by a "5" also has the five blue hexes it needs. Erase the orange hex at the start of that line to reveal a "2", which has only an obvious choice for a second blue hex. Once it's marked, we also give its respective diagonal, headed by a "3", the final blue hex it needs. Erase the final hex from this bubble, as well.

Let's make two final steps here. In the top-right bubble, you've probably seen that {6} line header. Well, we can now solve it. We've eliminated three of the line's nine hexes, and three more are marked; so now, just mark the other three! Notice how this line criss-crosses with another diagonal, governed by a {2}. We just gave that line a blue hex at an endpoint; mark the next one in line, then clear the other cells.

We are just about 3/4 of the way done with this one now; here's where we're at so far:



We can now complete the -2- diagonal in the top-right bubble, as well. Marking it shows that it already has two blue hexes, which clears several more cells.

We'll now turn attention to the top-left bubble for the first time. Remember that line headers ignore gaps within the same line. Using that logic, highlight the bubble's final column, which is governed by a -2-. Notice that it extends through the bottom-left bubble. Since we have a blue hex within the line on top of that bubble, clear the next hex up the column. We uncover another "0"; this will open up the section considerably.

The "2" we just revealed above that "0" has only two obvious choices for blue hexes--coincidentally solving the -2- column, as well. Additionally, the "1" at the bottom-center of the bubble has only one hex it can claim. You know the next two hexes to eliminate if you watch the shared blue hexes we just marked.

We can now complete the diagonal headed by the -2- extending from the bottom-left corner of this bubble; we just gave it the second blue hex it needed. Clear the final orange hexes from this line.





We can't quite complete the bubble yet, but we can make two more obvious moves. Give the "3" in the second column the only two remaining hexes it can claim, which will also give the "2" in the center of the bubble the second blue hex it needs. Clear the hex from above the "2" to reveal a "3", which we can't solve yet.

Both the top-right and bottom-right bubbles have just about the same amount of work to do. We'll try the top-right one first. It's a little tricky; the "4" already has three blue hexes but has two orange hexes it could claim. However, notice that they fall into a diagonal that is also governed by a "4". This line already has three blue hexes. Since we know that one of the two hexes by the empty "4" cell has to be marked, the other two cells within the line are eliminated next.

We've just eliminated a hex from the "2" in the bubble's first column, leaving it only one hex it can capture--which will be shared by the "4" Mark this cell, then erase the final orange hex from the "4".

The "2" on the bottom-left edge of the "0" is in much the same position here; mark the only other hex it can claim, which will also give both the "3" in the center, and the "1" along the right edge, their required blue hexes. Now, erase the orange hexes directly below each.

To solve the "3" we just revealed, we need to focus on the diagonal in which it falls, which is governed by a "6". The line has six blue hexes already; this eliminates the cell on the bottom-left edge of the "3", revealing a "2". Here again, the "2" has only one obvious hex it can capture; marking it solves the "3", and the final hex of the bubble is erased.


We are almost done now. We'll work on the bottom-right bubble now. Interestingly, along its bottom-left edge, we have a "2" and a pair of empty "1" cells in identical positions to their counterparts along the bottom-right edge. They'll be solved in the same manner, too. The {2} that pops up now has a blue hex at an endpoint in its chain of three active hexes; just mark the center one and erase the third.

We'll reveal a "1" above the {2}, which naturally shares a blue hex with it. Clear the orange hex on its top-right edge, then mark the only hexes the "3" that is revealed can claim. This coincidentally also solves that -2- diagonal.





Only a few hexes left on the board to solve. We'll need help from the line headers to complete the puzzle. Start with the diagonal governed by a "4" farthest left on the top-left bubble; the line now has four blue hexes. Erasing the last few hexes in its line allows us to solve the top, middle and bottom-right bubbles.

In the central bubble, we finally clear a hex from that -3- that has dogged us the entire time, and we can now mark the only one left in the bubble. Additionally, we've solved another hex in the column governed by a "5" that extends from the top bubble. With now only five active hexes in that line, we just need to mark the orange hex at the top of the column, also finishing the top bubble.

In the bottom-right bubble, we revealed a "6" of all things. Marking the last hex it still needs gives a second blue hex to the adjacent "2". The hex above the "2" is now erased; the "1" that appears shares a blue hex with the "2", and the last cell of the bubble is cleared.




And now, only three hexes remain, all within the top-left bubble. The diagonal governed by a "6" gives us the solution. The line already has six blue hexes. Erasing the final hex from the line gives us a "2"; with only one obvious choice for its second blue hex, it is also shared with the adjacent "3". The final hex of the puzzle is then erased to clear the level.



This puzzle is comparatively easier than Puzzle 3-1, but even here, it's important to study the interconnectedness of the cells on the board, as well as of the various lines extending through the different bubbles this puzzle gives us. As we saw several times, hexes we marked in one affected those in another by virtue of the restrictions placed upon the lines shared between them. Any time there is a puzzle with multiple grids, it's important to use the line-marking feature to determine how one line might affect the cells in several of them.

With that, we can move on to Puzzle 3-3. We're almost halfway done with chapter 3--and almost halfway done with Hexcells Plus.

Hexes Earned for Completing This Puzzle: 15

The next puzzle shows us how many hexes it takes to knit a nice scarf for the winter!




Personally, I would like my scarf to be uniform in color. Orange with spots of black just doesn't do it for me. :-)

So the difficulty in this puzzle lies primarily in the more complex relationships established between the cells, and the use of line restrictions to work through those relationships. In truth, the difficulty is actually a little bit easier than the last couple of puzzles; it's more about taking the time to think about the status of the puzzle before making a move in haste.

If you've already studied the empty hexes, you probably already see how we're going to open the puzzle: By clearing all of the hexes from that "0" just down from the top-center. Any time you have a "0" or a "6", go ahead and deal with it immediately as it will often open additional avenues. Interestingly, we get four "?" cells, a -2-, and a "1". The position of the -2- gives it only the standard ring of three consecutive orange hexes we've been seeing. Solve it in the same manner, and you'll give a blue hex to the "1". Naturally, its two remaining hexes will be erased.

The "2" we uncovered on the top-left edge of the -2- will share the latter's two blue hexes, which erases two more cells. We actually reveal another -2- from this, which is solved in the same manner as the first. Now, the "3" we just revealed at the top of the column governed by a "2" has only an obvious choice for its third blue hex; mark it now.

Let's finish one more group. The "2" at the top of the column governed by a "3" has the two blue hexes it needs. When we clear its remaining two orange hexes, we actually eliminate a cell from the {2} a couple of spaces down. Now, its choice for a second consecutive blue hex becomes clear, and the four orange hexes it still possesses are now erased. Let's stop here for a second; here's what we have to this point:

So far, the puzzle has been all about using the relationships between the hexes on the grid to determine what can be marked or eliminated. Most of the puzzle is like this; we'll examine the trickier parts in more detail when we get to them. For now, though, some cleanup work as we still have a few moves open to us.

Near the top-left corner, we uncovered another -2- cell that can be solved as we did the others. The "3" which appears near the top of the first column will then have only an obvious choice for its third blue hex. After marking it, we can clear the cells from the "1" positioned below the -2- we just solved.

Back over to the right a few columns, we have a "3" with three blue hexes and one orange cell beneath it to be cleared. Finally, back over to the top-center of the grid is a "2" with two blue hexes and a final orange hex on its top-right edge to erase.




We've solved the top-left quadrant now, but our paths forward are blocked off by "?" cells. This is where the solution becomes a little less obvious.

It's not very clear at first, but that diagonal governed by a "5" at the top-left corner of the grid holds the answer. Amazingly, we've already given that line four blue hexes. Highlighting it also shows that the line passes through two orange hexes which are adjacent to a "1" in the bottom-right quadrant. They also happen to be the only orange hexes the "1" has to work with, guaranteeing that whichever hex is claimed by that cell will also complete the line. So we can erase everything else along this diagonal, leaving only this pair of orange hexes. We'll get another "0" from this, so go ahead and erase everything around it.

Of the empty hexes we've just revealed, we can only work with one of them. Start from the empty "1" cell we were just looking at. Now, go two columns to the left and up two cells. This particular "2" has only two active hexes to work with; mark them next. Both of these blue hexes are shared with the "2" positioned along the bottom-right edge of this "2". That erases two more hexes, including one of the cells adjacent to the "1". We can now mark the final hex of that diagonal.

So now, the "1" positioned below the "0" we revealed a few moments ago has a blue hex, erasing the two orange cells still along its bottom edge. We'll get another "1" below this one, which shares the same blue hex. So the two final orange cells in this cluster can now be erased, as well.

Now, we revealed another "1" on top of the "1" we just solved to finish the diagonal headed by a "5". Each of them shares the same blue hex, so when we erase the orange hex from the upper "1", we also erase one from the -2- we revealed earlier. This leaves the -2- with only three total active hexes, with an endpoint already marked. We'll solve it like the others.

We reveal two more empty "2" cells from this, both of which have their required blue hexes; this clears a total of five more cells.

Continuing towards the right edge, the "3" near the bottom of the third column from the right has only two obvious blue hexes to claim, which will also give the -2- we just revealed a second blue hex. Clear that cell's remaining orange hexes. The "1" revealed above the -2- naturally shares one of the blue hexes, which erases two more cells.

Here's an interesting section. The final column is governed by a {3}. At the bottom, in the next column over, we have a "2" that needs a second blue hex, but both of its possibilities exist within the final column. So we know that the column's chain of blue hexes is going to start somewhere around here.

If we assume that the orange hex on the top-right edge of the "2" is the correct choice, the chain of blue hexes will extend upward to the third hex from the top; if the hex on the bottom-right edge of the "2" is the correct choice, the chain will simply reach the bottom of the column. Right away, this tells us that the top two hexes of the column have to be erased.

We reveal two empty "1" cells; however, the one at the very top has only one choice for a blue hex. Marking it clears the two hexes from the lower "1". Now, we get another "0" to clear two more cells. We're now left with only four active hexes in the column; this reveals that the bottom three hexes in the column are the ones to mark, and the last one in the column will be erased.

We're very nearly done with this puzzle now. Here's what we've completed to this point:



We've now left that empty "5" cell only five active hexes, so let's fill in the rest now. The columns governed by a "3" and a "4" on this side now have their required blue hexes; just be sure to erase the lone orange hex from the latter. When we do this, a "2" pops up with two surrounding blue hexes. Clear the one on its top-left edge.

We now reveal a -2- with a blue cell at an endpoint in its ring of active cells. Clear the next one in the line; another "2". This one also has two blue hexes. Clear the three which remain around it, and we can now solve the -2-.

This unveils another pair of empty "1" cells, the bottom of which gains its blue hex from the -2-. That leads to two more erasures and leaves only an obvious blue hex for the upper "1" to claim. Down the same column from the cell we mark is a pair of empty "2" cells, with a "1" two cells below them. That "1" already has a blue hex, so clear the hex between it and the empty "2" cells.

Now, to solve the empty "2" cells, as well as the column governed by a "1", we just need to give a shared hex which will complete both empty "2" cells at once. Can you choose the right one?






That's 3/4 of the puzzle complete now. Only the bottom-left quadrant remains. We have absolutely no information in this section except for that empty "4" cell at the bottom. There's certainly not enough information to complete the columns which run through this section.

This is the other part of the puzzle that can be pretty tricky to start. The answer lies in the diagonal governed by a "4", which extends from the upper-right corner. Highlighting it reveals that we already have three blue hexes. Notice, however, that the line extends through the "4" and two of its five choices for blue hexes. Since the line only needs one blue hex to complete it, we know that only one of those cells can be marked--which guarantees that the other three hexes around the "4" have to be marked.

This gives us an important new clue, as we have now given a third blue hex to the column within which the "4" lies. As this column is governed by a "3", clear its two remaining orange hexes. The "1" that pops up already has a blue hex, so this will clear four additional cells. The "1" that pops up on the top-left edge of the first one has only one choice for its lone blue hex, which will also give a third hex to the "3" we just revealed below it. Erase its remaining orange hex.

We just revealed another "3", which happens to fall into a column governed by a "3". The column now has only three active hexes; marking the last one also gives this new empty "3" cell the third blue hex it needs, so now, we can clear the two hexes to its left in the first column. The pair of empty "2" cells which appears will have only obvious hexes they can claim to complete this section.

This still leaves us with some uncertainty regarding the empty "4" cell. We still have one last column to work, which is governed by a "2"; it already has one blue hex. What's our simplest move within this column? Notice that in the column to its left, there is an empty "1" cell with a ring of four other empty hexes around it; the only two active hexes it can claim come from this column. This immediately tells us that whichever hex the "1" claims will also complete this column. So the two hexes below this pair have to be erased.

That reveals a "2" and another "1". The position of the "1" here gives us our next clue. On its upper-left edge is another "2"; notice that whichever hex that the "2" captures will also be shared with the "1". This means that the "1" cannot claim the cells on its upper-right or lower-right edges. Clear them next.



The -3- we just revealed is very important; since it only has the standard ring of four active hexes around it, we can go ahead and mark the endpoints. Doing so actually gives the column headed by a "2" the second blue hex it needs, and we can clear the column's final hex. Just as importantly, the hex we just marked is the second one for the "2" on its bottom-left edge. When we erase the final hex from the "2", we also leave the empty "4" cell with only four active hexes. When we mark the last one, we can also solve the diagonal by erasing its last orange hex. Finally, fill in the only hex remaining for that -3- to claim.

We're down to the final two hexes, with only one left to be marked. Start at that empty "4" cell we just solved, then go two spaces to the right. This particular empty "2" cell has only two active hexes to work with; mark the one it still needs, then clear the final hex to complete the puzzle.



Most of this puzzle involves establishing and using the relationships on the grid. However, as we have seen in these last several steps, it's also necessary to combine the information given to us by the line headers with the conditions on the grid to work out patterns that are either guaranteed to work or guaranteed not to work. We'll see more of this type of problem-solving throughout the rest of the game.

Hexes Earned for Completing This Puzzle: 15

Chapter 4 will now unlock if you would like to preview the next series of challenges--as well as the final puzzle mechanic that we'll be introduced to. Make sure to start with Puzzle 4-1 if you choose to do so; it introduces the mechanic with an explanation and fairly straightforward applications to it. We will go through it in tremendous detail when we get there in this walkthrough.




Another giant hex with some gaps and a bunch of embedded lines. Oh, boy.

First thing's first: Eliminate the cells by the zeroes on the outer edges of the grid.

The bottom gives us another "0" to work with; let's clear its hexes next. We even get another "0" in doing so, clearing two more cells.

Now, the upper-left and upper-right corners each give us a -2- cell with our normal ring of three continuous hexes. We know how to solve those by now, so get cracking. :-)

The bottom-right diagonal of the grid is headed by a {3}; we've left it only four active hexes, which tells us the two center orange hexes have to be marked. They'll be included in either possible combination of three consecutive blue hexes. One of the blue hexes we just marked falls into a column governed by a -2-, meaning we need to clear the next hex up that column. The "2" that this reveals is given the two blue hexes we just marked, meaning its four orange hexes are now also erased. That gives us the following to start with:



That gives us a good start to this puzzle. We can actually finish the column we're in, as well; notice that we've left only one unsolved active hex, which now must be marked. We'll now clear the orange hexes from the "1" below the hex we just marked. Next, we can refocus on the -2- we revealed when we were clearing hexes from the "2" a little bit ago. The last step left the -2- with only three consecutive active hexes, so solve it as usual.

Now, we can solve these bottom two diagonals, as well. The bottom one, governed by the {3}, now contains only three active hexes, so mark the last one. The line above it, headed by a "1", just got a blue hex from the -2- cell. So now, just erase the last hex in the line.




Near the right edge, the "2" that we revealed above the -2- we just solved now has two blue hexes. Eliminate the two orange hexes still bordering it. The "1" revealed in the final main column will share a blue hex with the "2". Erasing the cell above the "1" gives us yet another "2"; both it and the "2" on its bottom-left edge will now share the lone obvious hex they can each claim for their second blue one.

Now, in the diagonal governed by a "1" that we just solved, the empty "2" cell about halfway down the line gets its two blue hexes from the line below it. Clear the cell on its upper-left edge. This leaves the "1" right above the "2" with only one final choice for a blue hex. Mark it to give a shared blue hex to the "3" right above the "1".

The "1" directly below the "?" that we just revealed also has its required blue hex. Erasing the cell on its upper-left edge leaves this embedded column, governed by a "1", with only one active hex for us to mark. It also leaves the "1" on the upper-left edge of the "0" at the bottom of this column with only one obvious blue hex to claim. The "2" we just uncovered gains its two blue hexes from those we just marked; clear the last one from its ring.

We'll continue working our way clockwise around the bottom. The "1" above the pair of zeroes in the middle column has its blue hex; erasing the orange hex to its left leaves the "1" on the upper-left edge of the "0" with only one obvious hex to claim. Marking that gives a second blue hex to the "2" we just revealed above the "1". We'll erase the two orange hexes rimming the "2", then mark the only remaining active hex in this embedded column, also governed by a "1".

This has all left the "4" in the center column with only four total active hexes. We just need to mark the last one. To find our next move, however, we want to look to the diagonal governed by a "5" extending from the right side of the grid.

Interestingly, we've already given this line four blue hexes; it needs only one more. We can see that two of its remaining orange hexes border an empty "3" cell, which we had to leave earlier. They're also the only ones from which this particular "3" can claim its third blue hex, meaning one of them has to be marked--which, of course, would also complete the line's set of blue hexes. That means that the orange hexes within the line outside of this pair have to be erased:

So we've left the empty "2" cell nearest the bottom-left corner with only one remaining hex to capture, and it is shared with the empty "1" cells to the left. When we erase the orange hexes from the empty "1" cells, we'll leave only an obvious hex for the "2" above the one we just solved to claim.

When we get to this point, we want to solve the bottom-left diagonal, governed by a -3-. It has only three active cells now, with two already marked. After marking this particular hex, see if you can come up with the next series on your own:





Hopefully, you caught on to the fact that you have to complete the other embedded line in this section to complete that last sequence; it's really just a matter of watching the shared blue hexes to get to that empty "4" cell next to that column.

We get to finish two more lines immediately now. From the left side of the grid are diagonals headed by a "2" and a "6", each of which has the blue hexes they require. Erase their remaining orange hexes. We'll leave the "3" on the upper-left edge of the "4" we completed in the last sequence with only one final hex to claim; the "3" will share it with the "2" on its bottom-left edge, letting us erase the orange hex in the first column below the "?".

Now, let's focus on the column governed by a "2" near the left side of the grid; we've already solved its embedded line, but we need a blue hex from the top segment. Well, two of these three orange hexes border the "3" near the top of the second column. We can immediately see that one of them has to be marked, which eliminates the top hex of the column. The "1" thus revealed lets us clear away three more hexes, and we'll mark the final hex in this segment to complete the column and the blue hex requirements for the empty "3" cells to the left. Don't forget to clear the final hex from the first column!

Continuing to the right, then, the "2" we revealed on the upper-right edge of the "1" has only two active hexes to claim. The "2" below the "1" already has two blue hexes. Erasing the cell on its bottom-right edge gives us yet another "2". However, it gives the "2" located directly above it only one remaining obvious hex to claim, as well. That one will also give the "2" we just uncovered the second blue hex it needs, clearing two more cells:

We're almost done with Puzzle 3-4 now; here's where we left off:



We're definitely dealing with dual personalities at this point! We want to find the "2" located two cells above the "4" we solved earlier, just below where we are now; it already has two blue hexes, which lets us eliminate a vital cell from its bottom-right edge. Why is it vital? Because it lets us now solve the corresponding diagonal, governed by a "5", by marking the remaining hexes within it; there are now only five active hexes to worry about.

Marking these hexes gives the "3" we just revealed the third blue hex it needs. Erasing its remaining orange hex leaves the "2" right above it with only one hex to claim. When we mark it, we also give this entire column, which itself is governed by a "3", the blue hexes it needs. We can now erase the rest of this column.



The next move is not as obvious; those "?" cells we just revealed sort of stopped our momentum into the top-right section of the grid. If we scan the line headers, there's only one with which we can do anything. We have a diagonal governed by a "5" near the top-right corner, which already has all five of its blue hexes. Just seeing this alone lets us erase five more.

So let's start with the "1" we revealed near the beginning of the line; it naturally claims the blue hex marking the line's endpoint, so we'll erase the three orange hexes around it. The "2" next in line from the "1" now has only two obvious hexes to claim.

Let's turn attention now to the next diagonal above this one; it is governed by a -4- and now has only four total active hexes. We just need to mark the two which remain. Now, returning to the diagonal we just came from, we can see that the "3" next in line from the "2" we just worked now has three blue hexes. Erasing the remaining orange hex below it then lets us give an obvious blue hex to the next "3" in the line--coincidentally also solving that blue hex's respective diagonal, as well.

We have enough information to complete several more lines now. From the top-left corner, the diagonal headed by a "3" needs to claim its final blue hex by marking the last unsolved active cell in the line; and the line governed by a "5" right next to it already has five blue hexes, so we need to clear the last one. This solves those two orange hexes in the diagonal governed by a "5" from the right edge of the puzzle.

Lastly, the column governed by a "3" just to the right of the center has the three blue hexes it needs, erasing the three cells at the top of the column:



The criss-crossing diagonals governed by a "4" at the top of the grid are each now reduced to four active hexes apiece. Mark those that they each need.

We now can solve the center column; the "1" at the top has only one obvious choice for a blue hex. Mark that one, then erase the last hex in the line.

This leaves only two final orange cells in the top-right corner. The "1" here already has a blue hex, which will erase one of them. Mark the final hex to clear the level.



Honestly, there's nothing exceptionally tricky about this level once you get it started. Even though there are lines sub-divided with embedded lines, the way the puzzle plays almost completely eliminates this as even a factor to consider. It does take careful observation of shared hexes on the grid, and we certainly have to pay attention to the line restrictions, but on the whole, this puzzle is of only modest difficulty. It's still good practice for the true tests to come, however.

Hexes Earned for Completing This Puzzle: 16


ACK, A SNAKE!!!!!!!!!!!!!!!!



Ahem. Only a few active cells and lots of line headers make this puzzle appear confusing, but with the exception of a couple of tricky spots, it's actually not too bad.

As usual whenever we have a "0", we want to start by erasing its cells first to get a foothold into the broader pattern. When we do so here, we leave the "1" right above the "0" with only one possible hex to claim. As you might expect, that blue hex will be shared with the "1" at the top of the first column, letting us erase the hex at the top of the second column. The -2- that appears with a standard ring of three active hexes will be solved as normal.

We'll need to use line headers to progress any further. Notice that the first two of the three columns headed by a "4" here contain embedded lines. The embedded lines are each governed by a "2", meaning that for each, we need two blue hexes for the top segment and two blue hexes in the bottom segment.

Our biggest clue here is the empty "2" cell in the second column; it has only three active hexes around it, with one already marked. The other two on its right fall within the first of these columns. This segment of the column already has a blue hex; we can easily see that one of the hexes bordering the "2" has to be marked. So the bottom hex of this segment has to be eliminated.

The "2" that we uncover from this has only two obvious hexes to mark. From this, we'll also give the "2" that we just left its second blue hex, also giving this end of the column the second blue hex it needs. Go ahead and elminate the orange hex on the upper-right edge of the "2".

Now in the third column, we have a pair of empty "3" cells which each need a third blue hex. Their only choices come from the fourth column, which is laid out in similar fashion to the third--governed by a "4", with an embedded line headed by a "2". The upper segment just gained a blue hex from completion of that "2" at the bottom of the third column. So we can take a slightly different approach here. Since each "3" can get its third blue hex only from this column, and the top segment already has a blue hex, there is no choice but to mark the cell shared between each "3", then clear the other two hexes from this end of the line.

Since the "2" we just revealed at the top has its two blue hexes, the top two hexes of the next column can now be erased. The "1" revealed on top has only one possible hex to claim; the "1" will share this blue hex with the "2" that we uncovered below it, giving it a second blue hex in the process. We'll clear the two remaining orange hexes from this particular "2"; the "3" revealed below it also has only obvious hexes to claim.

Marking the two remaining hexes the "3" needed also gave the "2" we just revealed in the next column a second blue hex. So now, clear the two on its right. We reveal another pair of empty "3" cells; the lower one will have only obvious hexes to mark. However, we don't have a clear way to complete the upper "3" yet.

So our last step for this sequence takes us to our first diagonal to be completed. From the upper-left corner is a diagonal governed by a "2", to which we have now given two blue hexes. Highlight the line and then erase the rest of its cells, over in the right-center section of the grid:

We haven't worked too long on the puzzle, but we've already accomplished a decent amount. Here's where we left off:



So we just got two -2- cells and a "1" from completing that line. The first -2- has only three active cells to deal with; two rim its top edge, one is below it. We know immediately that the cell below it has to be marked; of course, that feeds a shared blue hex to the -2- in the middle. We can erase the hex directly below the second -2- as a result. This reveals a -3-.

The -3- is placed so that it has its standard ring of four active hexes, so we know immediately to mark the endpoints of that ring. We'll give a blue hex to the "1" in this line from doing so, and we can erase the hex above it. The "2" that this uncovers claims two obvious blue hexes, one of which will complete both of the -2- cells. We'll now erase the hex above the first -2-, revealing a "1" which shares a blue hex with that -2-.

Go ahead and erase the remaining orange hexes from the "1". We'll uncover a {3} from doing this; it already has two blue hexes, one of which lies at an endpoint. Mark the next one in sequence to complete the chain, then erase the orange hex which remains on top of it.




We'll try to work our way back up and around to the top-center, then. We just revealed another -3-, again with a standard ring of four continuous active hexes. As before, mark the other endpoint. We'll give a blue hex to the "1" we just uncovered from this. When we erase the hex on its upper-left edge, the "1" we uncover shares the same blue hex, clearing two more cells.

We reveal yet another "1", and another -2-. The -2- has four active hexes this time; erase the hex above the blue one that it owns. That reveals another -2- on the edge, which we can solve immediately. It has three active hexes, two of which lie together on its lower rim. Since one of those is marked, the second in the pair is erased, and we can mark the one on its upper-left edge.

By solving that -2- in this manner, we left the -3- located above the {3} with only three active hexes, so it will claim the last one in the corner. As a final step in this sequence, move back over to the -2- on the bottom-left edge of the -2- we just solved; it has two blue hexes now, so clear the last one on its top-left edge.



We'll go ahead and finish the final column of the puzzle now; it's governed by a "4" and already has the four blue hexes it needs. Go ahead and erase the orange hexes near the bottom of the column.

Returning to where we were, we have an empty "1" cell with only one hex it can claim. The "1" will share it with the "2" above, giving it a second blue hex and letting us clear two more cells. Each empty "1" cell revealed from this already has a blue hex; clear the three orange hexes surrounding them. Of the three additional empty "1" cells revealed from this, the bottom one already has a blue hex; clear the next one in line, then follow the "0" that appears.

Go ahead and follow the second "0" if you haven't already; the "3" uncovered above the first one will have only three obvious blue hexes to claim. Erase the hex above the "2" positioned on the upper-left edge of the second "0". The "4" will then claim the two obvious blue hexes it needs for a complete set. This also gives the "3" at the top of the column headed by the "2" its third blue hex, and the final hex in this section is cleared:

And just like that, we've solved about half of the puzzle. We need to return to the puzzle's final column and try to work our way clockwise to the end. We unveiled a pair of empty "1" cells at the bottom of that column, the top of which has a blue hex. We'll naturally clear the two hexes on its left, leaving the lower "1" with only one hex it can claim.

This sequence also leaves the -3- a few cells above with only a third obvious blue hex to mark; afterward, we can clear the next two hexes down that column, as the "2" and the "1" directly below the -3- already have their required blue hexes.

We reveal another -2- at the bottom now, with only a standard ring of three active hexes around it. An endpoint is already marked; just complete it as usual. Then, clear the hex on the bottom-left edge of the "2" we reveal from doing so.





The {2} we just revealed has a blue hex at an endpoint in its ring of active cells; mark the next one in line and clear the third one. The "1" we just uncovered shares the blue hex we just marked; eliminate the two cells to the left of the "1". This reveals another pair of empty "1" cells; the upper "1" shares the same blue hex, and when we erase its two orange hexes, the lower "1" will have only one hex to claim.

In fact, we'll solve the -2- that we just revealed as we have the others. The "1" revealed on top of the -2- will get a blue hex from this, clearing the cell on its upper-left edge. We reveal yet another "1" from this, sharing a blue hex with both the previous "1" and the -2-. Clear the hex on its lower-left edge.

We can now complete two diagonals. From the section of the grid we completed a few moments ago, we have diagonals governed by a "5" and a "3". For the line governed by a "5", we just reduced it to five active hexes; we now have four blue hexes to mark from this. Finally, for the line headed by a "3", it has only three active hexes now, with just one needing to be marked.

As a final step for this sequence, clear the remaining orange hexes from the nearby "3" and "2" as we have just completed their sets of blue hexes. Pay close attention here, and you'll find that we can clear quite a few more hexes from this, too!




We've got roughly 1/3 of the grid left to solve now. Under the "0" we revealed a moment ago is a "3" with only an obvious choice for its third blue hex. Go ahead and mark it. Notice now that the "3" itself lies in a column governed by a "5". Highlighting it shows that we have reduced it to five total active cells. So now, mark the two remaining in the center of the column.

The puzzle now ramps up somewhat in difficulty. We'll need to pay more attention to the line headers to finish. We do have enough information to solve another line. Extending from the upper-left corner is a diagonal governed by a "6". We have reduced it to six total active hexes, meaning we just need to mark its remaining unsolved hexes.

We get to do the same thing for the center column, which is governed by a "7". It now has seven total active hexes, so just mark those which remain.






In the center cluster now (the tail of the snake?), we've given one of the empty "4" cells four blue hexes; let's clear the two which remain around it.

Next, we'll solve three more lines. From the upper-right edge of the grid is a diagonal governed by a "3", to which we have given three blue hexes. This will erase several more cells. To the left of the center now are two more columns to solve. One, governed by a "4", simply needs its remaining hex to be colored; the next one is governed by a "2" and needs its final cell to be erased since it has two blue hexes.

We're at the end of Puzzle 3-5 now; here's the last sequence we completed:



Let's try to complete the central cluster now. The "4" at the left edge of the central cluster now has four blue hexes, which erases its last remaining orange hex. This leaves only an obvious blue hex both for the "2" below it, the newly-revealed "3", and the "5" to claim. This also completes the column governed by a "6" in this section.

We can complete the column governed by a "4" just to the right of center by marking the remaining active hexes; we've reduced it to only four. The "2" within this column already had two blue hexes. Erasing the hex on its bottom-right edge also leaves the nearby "4" with only one final blue hex to claim. For that matter, the "2" we just revealed only has two active hexes to work with, anyway. The "5" farther down now has its five blue hexes, so clear the last orange hex from it.

Well, the "?" we just revealed does us no favors, but if we focus on the intersecting lines, we can still finish this cluster. The diagonals in question all run from the upper-left edge of the grid and are headed by a "4", "5", and "6", respectively. All have their required blue hexes; erase the four remaining orange hexes in the central cluster.



We're left with only the bottom-left edge of the puzzle to complete. We want to complete the diagonal governed by a "5" from near the right-center of the grid next. Again, we've reduced the row to the same number of active hexes as the number in the header; mark the final hex at the end of the line.

This gives a second blue hex to the "2" positioned within this line, too. We can erase the orange hex on its upper-left edge. That reveals another "2", but it more importantly allows us to solve its respective diagonal, too. This one is governed by a "4" and again extends from near the right-center of the grid. The line now contains only four active hexes; mark the last one.

The "2" falling within this diagonal now has its second blue hex, as well. When we erase the orange hexes still rimming it, we'll leave the "1" on its upper-right edge with only one obvious blue hex to mark:





Nine orange hexes remain, with five blue ones to identify. This is the toughest part of the puzzle.

We're left with just a few remaining line headers, including the embedded lines we identified all the way back at the start. Each of the remaining empty hexes have multiple possibilities for their required blue hexes. What's our simplest move, then?

The simplest move can be identified by the column governed by a "4" which does not have an embedded line. The "3" within this column has four active hexes around it; the column has three of its four blue hexes. And of the empty "3" cell's three remaining choices for blue hexes, two fall into this column.

What this ultimately means is that we can't give the "3" a complete set of blue hexes without marking one from this column; since the line only needs one more blue hex itself, only one of these two can be selected. So we can identify the cell on the upper-left edge of the "3" as a guaranteed blue hex.

Notice what this did; it gave the blue hex's respective embedded line the second blue one it needed. That lets us clear a very important cell:







The "3" that this just revealed is positioned so that the hexes it claims are obvious; mark the two that it still needs. This also gives the "3" we've been working on the three blue hexes it needs! We can now erase the cell above it, thereby also completing the column governed by a "4".

Now, we can also complete our other embedded line. We've finally given it two blue hexes; erase the last one in the column.







We turn to the grid's final remaining line header for the closing sequence. Extending from near the upper-right corner, it is governed by a "7"; we have finally reduced the line to seven active hexes. We now mark the final hex in the line. This gives the "2" located directly below it the second blue hex it needs. Now, erase the hex below the "2" and mark the final hex to clear the puzzle.



This puzzle actually starts off relatively easy, and for about 2/3 of it, everything is pretty standard. It isn't until we reach the center cluster and then work back to the bottom-left edge that we have to approach things a little differently. But if you thought these last several puzzles were hard, trust me: We haven't even begun to sweat yet...

Hexes Earned for Completing This Puzzle: 16

This puzzle gives us one very small grid with a few empty cells and a lot of line restrictions.




Yowsa. How on Earth do we get this one started?

This is a puzzle that seems impossible to get going but which will ultimately unravel once we get the first few moves in. Our most important clues lie in the central column, governed by a "5", and the final column, governed by a "1".

In between these columns, we get a pair of empty "2" cells. What can we determine, then? Well, we know that of the four total blue cells we'll be marking for them, only one will come from that final column. This means three will come from the center column, making up three of the five hexes that column needs.

Back in Puzzle 2-6, we introduced a different method of counting up the hexes which can contribute to a particular line's blue hex count. We're going to use that here for a moment, too. Within the central column are four total hexes adjacent to the empty "2" cells; we know that three of the four have to be marked, so we're actually going to count that segment of the column as Blue Hexes 1, 2, and 3. Using this, we can guarantee that the column's top and bottom hexes have to be marked.



The hex we just marked at the top of the center column also gave its respective diagonal, which is governed by a "1", the lone blue hex it needs. We can now erase the top hex of the final column to complete the diagonal. Sadly, that only gives us a "?".

I'm not ashamed to admit that even for writing this guide, the next move that you can actually make logically took me a relative eternity to figure out. The trick is with the diagonal from the bottom-right corner, governed by a "2"; it stretches across two of the empty "2" cells. The "2" at the bottom-left corner received a blue hex from our opening; with the line containing only three active hexes, two of which border this "2", we can guarantee that it receives its second hex from this diagonal.

This alone gives us a few moves. First, we know that the hex at the start of the diagonal is a guaranteed blue hex. Secondly, since that blue hex also gives the final column the only one it needs, we can erase the rest of the column. Finally, it tells us that the hex on the upper-left edge of the "2" at the end of the diagonal has to be erased:


This gives us an important new sequence. The top empty "2" cell between the central and final columns now has only two active hexes to deal with; we can mark both of them within the center column now. When we do this, we also give a second blue cell to the "2" positioned between the first and center columns and can erase the two top hexes from the first column.

We've coincidentally solved the diagonal extending from the top-left corner that is governed by a "2". We can solve another diagonal, too; from the top-right corner is a line governed by a "1", and we just gave it a blue hex. Erase the one at the end of the line from the first column:




Two hexes left to mark, and four orange hexes to choose from. We gained an extremely useful empty "1" cell in the first column, with only one hex it can claim. Mark it to also give the column as a whole the only blue hex it needs, then clear the last one at the bottom of the column.

Interestingly, we also gave the diagonal governed by a "1" which extends from this column the lone blue hex it needs. This means we can erase that line's final hex and mark the final cell of the puzzle to clear Chapter 3!



Hexes Earned for Completing This Puzzle: 20

Congratulations for reaching the halfway mark in Hexcells Plus. At this point, we are introduced to one final puzzle mechanic. Make no mistake: This is a tough one to master.




Quick question: What do you notice about this puzzle that is different from every other puzzle we have ever worked to this point? I'll give you a minute.

...

Time's up. In each grid of this puzzle, we are given a blue hex with a number inside of it. This is the hub for a Grid Overlay. Grid Overlays are highlighted regions which extend across the puzzle board, encompassing everything within a two-hex radius. The number inside of the hub tells us the number of hexes within that highlight which must be marked blue. However: This only relates to surrounding hexes! The hub itself does not contribute to the blue hex count for an overlay!!!

You may see me refer to such hubs by a few different names, which are interchangeable. Such terms include: "Blue Hubs"; "Grid Hubs" or "Overlay Hubs"; "Numbered Blue Hexes"; or something like "Blue xx," where the "xx" represents the actual number inside of it, such as "Blue 8." This notation will be preferred when attempting to identify a specific overlay.

Now, these types of hexes still retain one very imporant property: They're still considered blue hexes! Just because they're numbered doesn't make them any less blue than a regular blue hex, which means they still contribute towards the requirements of any nearby empty hexes, or to the restrictions of any numbered lines into which they may fall. We'll talk about this more as we go along, too.

We'll go ahead and take the first grid, then. To extend the overlay from the hub, which in this case is governed by a "10", just click on the cell itself. You'll see a faint white highlight extend from the hub:



Great. Now how do we solve it? Well, this particular hub needs to have 10 total blue hexes surrounding it; remember that the hub itself does not count! So what do we find if we count the number of blue hexes within its radius which are already given to us?



One other thing to notice is that the overlay covers the entire first grid. The maximum number of hexes that any overlay can encompass is 18--the exact number of hexes surrounding the Blue 10 hub (The grid contains 19 total hexes, including the hub.). This will become important as you deal with numbered blue hexes requiring much higher numbers of blue hexes within their radii--such as the one we're getting ready to tackle.

Once you're done with an overlay, you can right-click on the hub to dim out the number, similar to how we have seen with line headers.

Now, let's move to the second grid. Go ahead and click on the Blue 12 to extend its overlay. We have six "?" cells surrounding the hub, and we need 12 blue hexes to fall within the scope of this overlay. With this overlay, we want to count the total number of orange hexes falling within its radius. Did you count 12?



You can probably guess the next step: We're going to mark all of the orange hexes within the radius of this overlay. After you're done, notice that the "REMAINING" counter now reads "0"; erase everything else to complete this puzzle.



Before we close this puzzle out, I want to show off one more thing. Notice that while we were marking the cells within the overlay that two additional Blue Hubs were revealed. For the rest of the game, expect that at least some of the blue hexes that you mark will actually become hubs which reveal additional Grid Overlays.

Let's look at one of them. Go ahead and dim out the one we just completed, then click on one of the Blue 5 hubs; either one is fine. I'm going to use the one on the left for the next image.

You'll see that the overlay encompasses a number of empty hexes, plus some of the cells we just marked. And that's the real trick with Grid Overlays; they tell us how many, but not necessarily which ones. The number only tells us the number of blue hexes the overlay will encompass, but as we will soon see, it's necessary to use the other restrictions on the grid to lead us to the ones we actually mark. In other words: Treat Grid Overlays as a tool to assist with completion of a grid. Always scrutinize the information they reveal; don't take them at face value without analyzing everything that influences their part of the grid first.



Hexes Earned for Completing This Puzzle: 5

The next challenge introduces us to actually using Grid Overlays in the course of normal puzzle-solving. However, it is a relatively gentle increase in difficulty and does not throw us into anything terribly complex. We'll discuss how to treat the overlays as we come to them.




It's easy to be drawn immediately to those numbered blue hexes, but as with any puzzle, we want to take in the whole picture before doing anything. Remember: Grid Overlays are just another tool to use to determine the proper solution; they don't take priority over anything else. In fact, this is the best way to approach them: As a resource to check when nothing else seems obvious.

That said, in the lower-left section of the grid are a "1", a "2", and a {2}. The "2" actually has only two possible blue hexes to claim, so start by marking them. Notice that when we do so, we feed a blue hex to the {2} at an endpoint in the surrounding cluster. So now, mark the next hex in line and erase the other two orange hexes. Additionally, we gave a blue hex to the "1", so we can clear its remaining cells, as well.

We revealed a pair of empty "1" cells when we cleared the cells from the {2}; notice that the "1" on its upper-right edge already has a blue hex from the {2}. We can clear the hex above the "1", which will leave the second "1", which is right above the {2}, only one obvious blue hex to claim. That blue hex will mark an endpoint in the ring of hexes around the {2} that we just uncovered, so we can now easily solve this one.

Now, on the left edge is another "2", but here again, there is only one obvious hex to give it a second blue one. We'll mark it, also giving a shared blue hex to the "2" we uncovered on the upper-right edge of the first. We'll erase the cell above that one next.




So far, we haven't even had to touch the numbered blue hexes. Continuing what we were doing, the "2" at the top of the second column already has two blue hexes, so we can erase the one on its upper-right edge. That reveals a "3", which has four total active hexes and no obvious guidance on which remaining orange hex it will claim. In the center of the grid is another "3" in much the same position.

This is a good time to go ahead and use our overlays. Let's start with the one at the bottom-center, with a "2" inside. Go ahead and click the cell to extend the overlay. Notice that only four cells are enveloped by this particular overlay. Remember that the "2" inside of the hub tells us that two cells within the overlay must be marked blue. We have this; two of the cells highlighted were among the first cells we marked. This means that the hub has a complete set of blue hexes; now, erase the other two cells within the highlight, then right-click on the hub to deactivate it.

We can also check the overlay at the upper-left corner of the grid. Extending it shows that five total cells are covered: One blue, two empty, and two orange. The overlay needs three total blue hexes. In this case, we have only two orange hexes encompassed by the overlay to choose from; go ahead and mark them.

When we do this, we give a third blue hex to the "3" positioned on that corner. So now, we can erase the remaining orange hex on its upper-right edge.






There's still some uncertainty over which cells the "4" we just uncovered, and the "3" in the middle, can claim. So let's work in the bottom-right section of the grid next. We uncovered another pair of empty "1" cells a little while ago. They each have only one choice for a blue hex, which will give the "2" we started with the two blue hexes it needs. Erase those which remain from the "2".

The "1" we uncovered on top of the "2" shares a blue hex we just marked, clearing three more cells. This brings us up in line with that "3" in the center, which we can now solve. First, though, solve the -2- in the usual manner. Upon doing so, just mark the last hex that the "3" can claim.




We're almost done with this puzzle now. This is a good time to check the overlay off the top-right section of the grid. Go ahead and click the Blue 4 to extend it. This one needs four total blue hexes and contains one. The only other cells it covers are a single empty hex and three orange ones. We can see that these three orange hexes now must be marked to give the hub its required blue hexes.

Let's see what that does to the grid. The "2" positioned on the bottom-right edge of the -2- in this section now has its second blue hex. When we erase its remaining orange hex, we leave the "2" below this one with only one more to claim.

Let's get rid of three more cells. The "3" on top of the -2- now has three blue hexes, erasing the one on its upper-left edge and revealing another "4". Both it and the one across to the left now have their required blue hexes, allowing us to clear the cells above each:



We've got one final overlay left. However, before we get to it, we have a few more hexes we can erase. Both of the empty "3" cells along the upper rim of the Blue 8 hub have their required blue hexes, allowing us to erase three more.

Two orange hexes remain; one needs to be marked. Click the final overlay; it will tell us exactly how to solve the puzzle.



Hexes Earned for Completing This Puzzle: 10

Congratulations on making it through Puzzle 4-2. It's not that difficult if you just take in all of the information before making a move. Remember to use the Grid Overlays in conjunction with the other rules governing the cells on the grid.


Puzzle 4-3 ratchets up the difficulty a little bit; you'll have to be a little more critical in your analysis of how the Grid Overlays play into the overall pattern.



It doesn't really matter which grid we start with; let's just take the bottom first. We have two overlays, both governed by a "3". We also get two empty hexes: a -3- and a {4}. Remember: the hubs count as blue hexes, meaning that each empty hex has one blue towards their count at the outset.

The {4} is probably the easiest one to start with. It has five total active hexes around it, but that also gives it only two possible patterns for it to have four continuous blue hexes. This means the middle three cells in its ring have to be marked. So we can go ahead and color the two orange cells in the middle of the ring to start with.

Let's now extend the overlay nearest the {4}. Remember that the hub itself does not count towards its blue hex count, but we just marked two blue hexes which do. This means that the overlay itself needs one more blue hex within its radius. Do you see how all of the cells surrounding the {4} are included within the overlay?

What this tells us, then, is that whichever remaining cell the {4} captures will also complete the blue hex requirements for the overlay. Our next step is then to erase the cells within the overlay which do not surround the {4}:





So now, we just use the hexes on the grid to make our next moves. The "2" we just revealed in the bottom-right corner has only one possible hex it can claim for its second blue one, and it will also complete the sets required by both the {4} cell and the overlay hub. We'll erase the final orange hex on the top-left edge of the {4}.

Now, we also revealed a "1" at the top of this cluster, which shares a blue hex with the {4}. Can you complete the rest of this section?







We actually opened up another hub here; we'll see how its overlay factors into everything later...

Let's take the -3- now. We actually just cleared a cell from its ring, leaving it only four active hexes to deal with. Remember: With that specific pattern, the two endpoints are always marked, without exception. Guess what? When we do so here, we complete its blue hex requirement outright!

So we now get rid of the remaining hex on the upper-left edge of the -3-. We can also clear the hex directly above the "2" in the very center of this grid; we just gave it a second blue hex. The "3" just revealed will very clearly claim the hex right above it. We reveal another blue hub from this; we'll get to it shortly, as its interaction with the one we revealed on its right is actually both interesting and helpful.

For now, though, let's see what we can do with the second Blue 3 we had at the puzzle's opening. Go ahead and expand the overlay; we can see that it encompasses all of the cells adjacent to the "3" we just opened on the upper-left edge of the -3-. With the -3- itself now being solved, the overlay has two of the three blue hexes it requires; we can immediately see that whichever hex the "3" ends up claiming will also complete the hub's blue hex requirement. So as before, our next move is to erase the orange hexes not linked to the empty hex we're trying to solve:

We can again use the empty hexes we revealed in making our next moves. Completing the "2" in the bottom-left corner gives the "3" the third blue hex it needs. As a result, we get a virtually identical sequence going up the left side of the grid as we did on the right side.

So at the top of this grid, we've now revealed three additional hubs. Go ahead and examine them; click on one of them, study the cells it encompasses, then click on the hub again to retract the overlay. Do this for all three; get a feel for how they each interact. If you want, you can even extend them all at once. You'll notice that cells shared by multiple overlays will lose much more color than those only affected by one overlay.

We're going to start with the hub governed by a "5" in the top-center; there's a reason for this, which you will see shortly. Go ahead and extend that one; notice how it only covers five total active hexes, only one of which is still orange. So go ahead and mark that cell in the upper grid.

Now, go ahead and extend the other overlays, each governed by a "2". Do you see how each one extends over two blue hexes we have already marked? We now need to erase the orange hexes within each Blue 2 overlay to complete them.





Sadly, the four "?" cells we uncovered just now don't do much for us, so we'll need to tackle the top part of the larger grid now. Here, we have only one overlay, with two more empty hexes: Another {3}, and a "1". The hub is governed by a "4".

The {3} is surrounded by only four active hexes, which makes it easy to see that, as with the {4} below, the center hexes within its ring must be marked. Go ahead and mark them now, then extend the overlay.

The overlay now encompasses two blue hexes, so it needs two more. We can see that every cell surrounding the {3}, and every cell surrounding the "1", is encompassed here. More cells are involved, but the logic is the same as before: The cells claimed by these empty hexes will also contribute to the overlay's blue hex requirements. So here, we need to erase any orange hexes outside both of their respective clusters:

We again let the empty hexes we just revealed guide our next series of steps. Complete the "3" in the middle by giving it the only remaining hex it can claim, erase the remaining orange hexes from all of the empty "1" cells, then give the {3} its last remaining hex. Can you carry out the next few steps on your own?



One more cluster, and we do get a brain teaser here. We're out of empty hexes and have just two remaining cells to mark out of all of these orange ones. Our only guidance is three final hubs. How do we use this to figure out what to mark?

The lesson here is in using overlapping grids to root out which cells have to be marked or eliminated. Let's start with the Blue 3 in the center; see how it encompasses this whole cluster? It at least has one blue hex already, though.

Now, check out each of the Blue 2 hubs individually; they each only cover six of the cluster's active hexes. We know two things from this: First, that for each of these overlays, we'll get two blue hexes; Secondly, that they will also go to the Blue 3 hub. This eliminates the orange hexes outside the radius of each Blue 2 overlay:



You may now see the solution: The two cells directly above the Blue 3 are enveloped by both Blue 2 overlays, as well as the Blue 3 overlay. Since these hexes are shared by all three overlays, they're also the only ones we can mark to fulfill all of their requirements. Mark them, then erase everything else to complete the puzzle.



Hexes Earned for Completing This Puzzle: 14

We are halfway through Chapter 4 now. Chapter 5 unlocks at this point. If you are still having some difficulty with Grid Overlays, I would recommend replaying the last few puzzles a few times; in order to fully understand the hardest puzzles, it's vital that you understand when to try to use them and when not to.

Also, you may have noticed that I have stopped explaining every single relationship between hexes on the grid as thoroughly as before. At this point, I'm assume that you have a fairly solid understanding of this and can, for example, pick out shared hexes between empty cells at least most of the time. The goal of this guide, as with my original Hexcells guide, is to help you become a better player, and if it's not, then it's worthless to you. Additionally, for those of you who are improving your skills and do understand the simpler patterns, I'm doing a disservice to you by treating you like you don't. Instead, I want to focus more time and attention on the logic of the harder sequences we'll begin encountering throughout the rest of the game. But that doesn't mean I'll stop pointing out simpler relationships altogether; sometimes, they're good reference points to larger sequences.

Okay, rant over now. I just want the guide to have the right "feel" to it, if you will. :-)


Puzzle 4-4 gives us a fairly long, very narrow grid with only a few empty hexes and three overlays to start. From a puzzle-solving standpoint, we'll begin uncovering many more additional overlay hubs as we fill in the blue hexes, and we'll need to refer to them more often to fill in missing information. We're also going to return to our modified counting scheme. Remember how we've started counting groups of cells as one unit with respect to lines of hexes? Well, beginning with this puzzle, we begin applying that scheme to Grid Overlays, too. Don't worry; there will be plenty of discussion on this as we go along.



The most obvious place to start is the top-right corner, where we have a {4} and a -3- fairly close together. The -3- has our standard ring of four orange hexes, so we know immediately to mark the endpoints. When we do this, we also give a blue hex to the {4} at an endpoint, making it easy to solve. As we mark the hexes that the {4} requires, we'll also give the -3- its third blue hex, letting us erase the cell directly below it.

Now, I want to show you something. I want you to deliberately make a mistake with me. Why? To show you something about Grid Overlays and why they can be deceptive. We're doing this now because when we restart the puzzle, it's no sweat to repeat that last step.

So what are we going to do? Do you see the {2} at the bottom, with the Blue 2 nearby on its right? Go ahead and click on that hub to extend its overlay; it covers two of the three active cells bordering the {2}. We know that the center hex of its ring has to be marked. Now, the easy thinking is, "Okay, I need one more hex around the {2}, and when I mark it, the hub is solved, too. So the hex to the right of the blue one has to be the one I mark, right?" Go on: Try and mark it.

ZAP!!!



"What happened?" you probably are asking, especially if you fell into that trap before accompanying me on this journey. It's definitely not hard to let an overlay seduce you into trying to do something that, when in the context of more evidence, simply falls apart. The reason this specific example falls apart is simply because of the fact that the overlay doesn't cover all of the possible hexes the {2} can capture; and even if it did, that would simply have us erase all orange hexes outside of that cluster. We would still need more evidence on the grid to determine which hex to mark besides the middle one. The moral, then, is simply to uncover as many clues as possible before trying to mark or erase a particular hex; often, especially when dealing with Grid Overlays, there is a piece or two of information that you need before you can make a guaranteed move.

Okay, experiment over. :-) Go ahead and restart the puzzle by clicking on Menu at the top-left corner of the screen, then simply clicking 4-4 in the Puzzle Index. Once the puzzle is reset, go ahead and solve those {4} and -3- cells again.

Now, for the real solution. When we marked those blue hexes, we actually opened up several more overlay hubs. When dealing with multiple overlays, it's best to try and start with the one that covers the least number of hexes first. In this case, it's the Blue 5 positioned on the top-right edge of the {4}. Expanding it shows that the hub covers only five total active hexes, with four already blue; to complete it, just mark its last orange hex. When we do this, we give the "3" located below the -3- a third blue hex, which erases two more cells from the grid:

So let's take the Blue 5 below the one we just solved and expand it. This one again needs five total blue hexes. If we examine it carefully, we find that it does have five blue hexes within its radius, with three additional orange cells included as well. In this case, then, we'll erase those three orange hexes to complete it.

What about the Blue 6 located below this hub, then? Expand it; this one covers a large number of hexes, including five blue ones. But where do we get Number 6?

Here's where we justify the mistake we made earlier. Like the Blue 2 we examined during our experiment, the Blue 6 overlay covers two of the three hexes bordering the {2} at the bottom. But we know that with any {2} surrounded only by a ring of three consecutive orange hexes that the middle cell of the ring will be marked. This now becomes the sixth blue hex owned by the Blue 6 hub, and the overlay's four remaining hexes will then be erased.

So now, let's do some cleanup work here. For the "1" on the top-left edge of the {4} from before, we need to erase its remaining orange hex since it shares a blue one with the {4}. The {2} at the bottom still needs its obvious second blue hex to be marked. Doing this gives a shared blue hex to the "2" positioned on the upper-left edge of the Blue 5 hub here, letting us clear two more cells from the "2". And finally, the overlay of that Blue 2 at the bottom-right now encompasses only two active hexes, so we need to mark that final orange hex right on the edge.

We're learning more and more about how to use Grid Overlays with this puzzle; here's where we're at:



We're at a place now where it's getting harder to determine what our next moves are. We can go ahead and at least examine the remaining overlays; it never hurts to look at one and see if it's helpful. Start with the Blue 5 on top of the {2} we just solved. Expanding it shows that it covers only five total active hexes, four of which are already blue. Go ahead and mark the last one it needs.

We can now expand the Blue 3 overlay right next to this one. However, if we look closely, it tells us nothing we don't already know; the nearby "3" has four total active hexes, with two already blue. The overlay covers these active hexes but no others, so we can't add or eliminate anything here.

The Blue 6 near the top, however, is useful for one more elimination. First, take the -3- on its top-left edge. We see immediately that it has its standard ring of four active cells, so we can go ahead and mark its second endpoint. Now, in expanding the overlay, we'll see that it covers five total blue hexes; it's obvious that whichever hex the -3- captures will also become the sixth blue hex that the hub requires. So we can erase the orange hex within the overlay which cannot contribute either to it or to the -3-:

We're at a logical dead end on this side of the puzzle now; there is no way for us to determine with 100% certainty which hexes will complete the -3- or even the "3" closer to the bottom. We'll need to work our way back here later.

We'll start with the {2} in the top-left corner; notice that it already has one of its blue hexes, which itself is a Blue 1 hub. This is actually fairly straightforward; we know that either the hex to the left or to the right of the hub itself has to be marked to give the {2} a second consecutive blue hex. So when we extend the overlay, our next step is to erase everything within it except for these two specific orange hexes.

When we erase these cells, the "2" which appears at the top of the first column has only two possible hexes to claim, which will give the {2} its second blue one. See if you can carry out the rest of this series:





Here's where we get to play with our new numbering scheme again. Go ahead and expand the overlay from the Blue 3 hub in the first column. Notice how it encompasses all of the active cells around the "2" on its bottom-right edge, as well as the last hex of the first column, and the blue hex at the top of the second column. We need two more blue hexes here; how do we determine which ones to mark?

We can figure out one of them by adapting our counting scheme to overlays. We know that one of the two hexes we mark has to come from the cells around the "2"; from this, we can see that the only other orange hex within the overlay is the one at the bottom of the first column. This one becomes a guaranteed blue hex.



This still gets us no closer than choosing a second blue hex for the "2"; something is still missing. Shift focus now to the Blue 2 hub at the bottom of the section; expand its overlay next.

This is interesting; this particular overlay encompasses the blue hex we just marked, as well as all active hexes around the "2" except for the Blue 3 hub. We've already established that one of the orange hexes around the "2" will be marked; in this case, that hex will also complete both of these overlays. So now, erase all of the orange hexes within this overlay except those surrounding the "2":

The relationships in this next sequence are just a little trickier. The empty hexes to focus on are: The "2" at the bottom of the second column; the "1" directly below the {2} we solved at the top of the third column; and the "2" on the bottom-right edge of this "1". You should be able to make it here now:



We're left with two remaining overlays on this side now, governed by a "2" and a "3". They actually require a similar approach to what we just did. First, go ahead and expand the Blue 3. The overlay already has two blue hexes; it also covers the "3" on the hub's upper-right edge, as well as all of its surrounding hexes. So the overlay's third blue hex will come from the one we give to the "3". As such, we can erase the one hex it covers which is not connected to the "3".

We take a nearly identical approach now with the Blue 2 overlay on top. It also covers the "3" and all of its surrounding hexes, including the Blue 3 hub. So do the same thing, and erase from this overlay those hexes not connected to the "3":





The -2- revealed here is particularly interesting; it has two orange hexes on its right, and only one on its left. Clearly, the one on its left has to be marked or else it will get two consecutive blues instead of two which are separated. This blue one gives the "3" the last one it needs. We'll ultimately eliminate two more hexes from this, and both of our hubs are solved.

But we opened up another hub yet from this, too, which is governed by a "4". What does it tell us?








The Blue 4 overlay is a bit trickier. Expanding it shows that it covers both the "2" and -2- we revealed a moment ago, and all of their active hexes. It contains two blue hexes already; we need two more.

It's best to run through the possibilities here, which is easy since there are only a couple. If we mark the middle orange hex here, we actually see that the hexes above and below it have to be erased, making it impossible to mark a fourth blue hex within the overlay. This means the top and bottom orange hexes here have to now be marked. Once this is done, we'll get several more erasures and one more obvious hex to mark:

At the top-center is an empty "2" cell; we can go ahead and mark the obvious second blue hex it claims. Now, directly below it is a Blue 5 hub, which when expanded covers three blue hexes and all of the active cells of the "3" on its bottom-right edge. We know that one of the orange hexes bordering the "3" will become the fourth blue hex for the hub. That leaves only one remaining orange hex within the overlay, which is guaranteed to become its fifth one. Mark it next.

We can use a similar approach with the Blue 1 at the bottom that we revealed a moment ago. When it's expanded, the overlay covers all of the active hexes of the "2" on its upper-right edge. The hub itself clearly gives the "2" its first blue hex; whichever hex we mark for its second will also give the hub the one it needs. So the remaining orange hex covered by the overlay but not connected to the "2" now has to be erased.

We're nearing the end of this relatively complex solution. Here's the last sequence we completed:



The {2} that we just uncovered gives us a very interesting solution. We already know that since it has our standard ring of three consecutive active hexes surrounding it, the middle one has to be marked. When we do so, we give both the "2" and the Blue 1 hub their remaining required blue hexes. Since the hex on the top-left edge of the {2} falls within the overlay, that hex has to be erased, and the one on its top-right edge becomes its second consecutive blue hex.

So another good time to study a particular overlay is when the number of required blue hexes is low. That said, let's take the Blue 2 we just revealed. When we expand it, we find that it covers one blue hex already; it also covers a large number of orange hexes, including those of the "3" positioned two cells above the {2} we just solved.

Examining the "3", we find that it has two blue hexes already; the two orange hexes on its right, which are covered by the overlay, are the only ones it can claim. So again, whichever hex we mark here will also complete the blue hex requirements of the overlay we want to solve. This lets us erase the five additional orange hexes this overlay covers.


These eliminations leave us with only a few remaining orange hexes. We can go back to using relationships on the grid now to make our next series of moves. Start with the "3" located on the upper-left edge of the Blue 3 hub we had to abandon a long time ago, and see if you can get this far:



So as not to get too far ahead, I didn't complete a couple more moves that are now open to us. We should now be able to complete the puzzle, however. To the left of the "0" we uncovered a few minutes ago is a series of three empty "1" cells going up that column, with a "2" above them. That "2" has only two obvious blue hexes to claim. Marking them sets up the puzzle's solution:



This puzzle is reasonably difficult, in large part due to the sheer number of overlays that we end up working with. At the beginning, there are only a few; however, as we fill it in further, we open up a significant number of additional overlays. However, we don't have to solve all of them; that is to say, we don't necessarily have to open each individual overlay we discover. Often, the conditions on the grid, including the overlays which we do solve directly, are enough to complete the extras. In fact, it can be a good idea to strike out any and all overlays which are solved, either directly or indirectly, just to eliminate clutter and confusion. In a puzzle relatively small such as this, that's not quite as important, but in large puzzles, this can be very helpful in cleaning up completed areas.

Hexes Earned for Completing This Puzzle: 15

Puzzle 4-5 more or less just builds upon what we learned in Puzzle 4-4:




It's a small puzzle, but it is challenging. Initially, our biggest problem is in determining which of the Grid Overlays to start with. It would seem that the Blue 7 might make the most sense, since it covers all of the empty cells in the region and all of their active cells. However, there's one glaring problem: Because some of the orange hexes are shared between the empty cells, there's no 100% sure way to determine how many of them will be marked, and how many will be marked which do not border the empty hexes.

So while it may not make sense initially, we actually want to expand the Blue 2 overlay first. It covers the empty "4" cell and three of its active hexes. How does that help us, though? Well, the "4" has five total active hexes; one of them is blue, the others are orange. Three of its four orange hexes are covered by the overlay. What we find, then, is that it is impossible to give the "4" the blue hexes it needs without marking two--but only two--within the overlay; as such, the hex to the right of its blue one has to be marked. This also means that every hex within this overlay not adjacent to the "4" has to be erased:

This leads us to a few obvious blue hexes we can mark. Notice that the empty "2" cells on the bottom-left and upper-right edges of the Blue 2 hub each have only one choice for their second blue hexes. Also, near the upper-left and upper-right edges of this part of the grid, the empty "1" cells have only obvious blue hexes to claim, as well. Marking them lets us solve a pretty large number of cells by just examining the relationships on the grid:

And just like that, almost half of this puzzle is solved! None of that last sequence even uses the Grid Overlays uncovered during its completion. However, we will want to use them going forward.

Starting near the upper-right edge of this section, expand the Blue 2 overlay that we uncovered earlier. It already encompasses two blue hexes, which will erase five more cells.

Up in the outer ring of the grid, go ahead and clear the cell by the "0" that we uncover, then mark the cells that the empty "1" cells in the same area need. We'll be able to solve a decent chunk of the outer ring from this:





The "1" at the top-right corner of the main section has only one hex it can claim, so mark it next. We'll need to return to the additional overlays to continue, though.

To the left of the Blue 7 hub we were given in the beginning is a pair of Blue 5 hubs. Start with the upper one. When expanded, it covers four blue hexes already, and only three orange hexes. Two of those orange hexes border an empty "3" cell which still needs a final blue hex; these are also the only ones from which the "3" can get that third blue hex. This eliminates the orange hex positioned below the lower Blue 5 hub.

We'll get another elimination from the Blue 7 in the center. Its overlay covers six of the seven blue hexes it needs; it's not hard to see that the seventh will come from whichever hex we give to the empty "3" cell below the hub. So the orange hex it covers which is not connected to the "3" has to be erased.

Now, expand the Blue 6 overlay on the upper-right edge of the Blue 7. This one covers four blue and three orange hexes. Two of the orange hexes belong to the "2" on the hub's bottom-right edge. As the hub itself contributes its first blue hex, we know that only one of its two orange ones can be marked. That leaves the third orange hex, positioned on the bottom-right edge of the "3" in the middle, as a guaranteed blue hex. This gives that "3" its third blue hex, but don't go any further after marking it; we'll clean this up next.



We just gave the "3" in the middle, below the Blue 7 hub, its complete set of blue hexes; erase the two orange ones which remain around it. Just from this, we'll get one more guaranteed blue hex to complete the "2" on the bottom-right edge of the Blue 6. This, in turn, lets us erase several more hexes:

We are quickly approaching the end of Chapter 4 and have now completed most of Puzzle 4-5. Here's where we're at:



Let's expand the Blue 5 hub on the bottom-right edge of the "3" we just solved; it covers four blue hexes and three orange hexes around the outer edge of the cluster. These orange hexes happen to run adjacent to a series of three empty "2" cells. Interestingly, the first two, which are positioned on the bottom-center and bottom-right edges of the hub, each have two options for their second blue hex. But since we can only mark one, our only choice is to mark the hex shared between the empty "2" cells bordering the Blue 5, then erase the other two. From this, we can complete the main part of the grid and some of the outer ring just by using the relationships between the cells:

This leaves only about half of the outer ring left to complete. We'll need to start using the overlays which connect the ring to the main part of the grid. Our best place to start is with the Blue 7 overlay at the bottom-right corner of the main cluster. When we expand it, it covers exactly seven active hexes. Just mark the last few.

Now, expand the Blue 3 overlay a few cells down from the Blue 7. We just marked the third blue hex it needs, so we can now erase the two orange hexes left within its radius. For the "1" revealed near the bottom-center, go ahead and mark its obvious blue hex, as well.




We still don't have enough information to solve the "2" at the bottom-center. Let's move back to the left-center of the main area. The "2" in the cluster connecting to the outer ring has its two blue hexes, so erase the orange one on its bottom-left edge. That reveals a "3"; unfortunately, its third blue hex is not yet clear.

We can use the Blue 5 overlay on the bottom-right edge of the "3" to aid us. When expanded, it covers four blue hexes, as well as the "3", all of its active hexes, and one extraneous orange hex. We know that the blue hex which completes the "3" will also complete the hub's blue hex requirements. Erase the orange hex which is not connected to the "3", and then we'll be able to solve this small section:

We're down to the last few cells. To solve the top-left corner, just expand that Blue 5 overlay; it now encompasses only five total active hexes, so mark the last two in that corner.

To start the bottom-left corner now, expand the Blue 3 overlay we just uncovered; it covers only three total active hexes, letting us mark the first two cells going down that line.






Only four hexes left, with only one blue hex to mark. We can eliminate half of them immediately; since two of them border that "2" at the bottom-center, we know that one will become the puzzle's final blue hex. So erase the two hexes not connected to the "2".

The "1" that we uncover adjacent to the orange hex on the bottom-left edge of the "2" tells us that this is the cell we have to mark; do this, then erase the final hex to complete the puzzle.



Hexes Earned for Completing This Puzzle: 18

And with that, we're down to the final puzzle of Chapter 4. With this one, we introduce a new wrinkle to Grid Overlays:


Hmm...Apparently, even overlay hubs can have either consecutive or disjointed blue hexes within their clusters...



Stylistically, this is one of the most interesting puzzles in the game: A series of interconnected rings, each with a different Blue Hub in the center. Each hub extends an overlay which exactly encompasses the ring in which it is housed; it covers no other cells.

Naturally, since the rings are interconnected, there is some sharing of cells between different overlays, and this is where the challenge lies. We're given precious little information to determine which cells to mark or to exclude.

Probably the easiest way to refer to each ring will be by the Blue Hubs in their center. There are actually three represented by a Blue 5 hub, but they are in different rows; for those, I'll also give the appropriate row, numbered from top to bottom. The very top and bottom rings, being alone, won't be given row numbers; only the middle sections containing multiple rings will be considered as being rows for purposes of this walkthrough.

We actually want to start with the top and bottom rings. We'll begin at the bottom, with a Blue {5} cell in the middle. We're already given a blue hex at the top-center of the ring; we know that the five blue hexes ultimately marked have to be consecutive within the ring. Go ahead and expand the overlay; you'll see that no hexes outside of the ring are highlighted. If we take the blue hex as an endpoint, we can determine the maximum reach of the chain of blue hexes going to the left and to the right. When we do so, we find that the three bottom-center hexes cannot be marked. So start by erasing all of them. We'll get two zeroes to clear away a couple more hexes, too.

With that done, we leave only seven active hexes in the ring. If we count out the possible combinations like we would for a line of seven hexes governed by a {5}, we find that the center three hexes have to be marked. So now, mark the hexes on either side of the blue one we're given.

That's all we can do down here for now; we can take the top-center ring now, which has a Blue 10 for a hub. There are actually 12 total hexes in all of the rings, in this case including the empty "1" at the bottom-center. This "1" has three active cells around it. Three of the ring's total hexes include the "1" and two of its adjacent orange hexes. Subtract those three, and we have nine total hexes. We can determine three things from this: 1) That these nine hexes are guaranteed to be marked; 2) That the "1" will claim either the hex to its left or its right; and 3) That the "1" cannot capture the hex beneath it, since it would then be impossible to mark 10 blue hexes for the hub. So first, mark the nine hexes we've just guaranteed, and then erase the hex directly below the "1":

The "1" that we just revealed from the erasure has only one possible blue hex to claim; mark it next. Notice how one of the rings in which this hex falls has a Blue 3 in the middle; the one to the left of it has another Blue 10.

Go ahead and expand each of these overlays; we find that three orange hexes forming the boundary between these rings are shared by each overlay. This gives us another nine obvious blue hexes for the Blue 10; we can't give it 10 blue hexes without marking one of these three, and the conditions now governing the Blue 3 in the next ring dictate that only one of them can be marked. So mark the nine obvious hexes; afterward, erase all of the remaining hexes in the Blue 3's ring except for the three in question:

It's rings across America! We've cracked the surface but have awhile to go; here's our last sequence:



The "1" we revealed at the top of the Blue 3 ring helps us here as it already has a blue hex. We'll erase the orange hex on its bottom-left edge. The "2" that this reveals tells us everything we need to know to complete this section. We can also finish off the top-center ring; we've left it with only one orange hex, which now must be marked to complete the Blue 10 hub.

The first ring in the middle row has a Blue {4} in the middle, and we have now given this ring two consecutive blue hexes. Since one of them is at an endpoint given our last step, we just mark the next two in sequence and erase the rest of the ring to complete it. And as a bonus? If we follow the "0" that we uncover here, it allows us to complete the first ring of the third row, too!



Let's take the very center ring now, which has a Blue 5 hub in the center. Expanding the overlay shows that it covers only five total active cells; naturally, we want to mark the remaining unsolved hexes here next. When we do this, we also give a pair of blue hexes to each of the empty "2" cells along the ring's right edge, letting us erase the two orange hexes adjacent to them. We'll reveal two more zeroes from this, letting us erase two more still.

We're left with an interesting situation concerning the final ring of the top row, which has a Blue {2} in it. The ring's bottom-left hex is a "1", and its only two active hexes are in this ring. This instantly tells us that the ring's pair of consecutive blue hexes will start from whichever hex the "1" ultimately claims, and as such, we can determine the four possible hexes which satisfy these conditions. So every other cell within the ring can now be erased:

Now, the Blue 5 Hub in the ring to the left establishes the conditions for solving this one. The two orange hexes above the "1" naturally fall into both rings; if we were to mark the hex directly above the "1", the Blue {2} hub tells us that we would automatically have to mark the next hex up. And in doing this, we would give the Blue 5 Hub six blue hexes. So clearly, we have to erase the two orange hexes above the "1", then mark the last two hexes within the Blue {2} ring. That leaves only one final hex to mark to complete the Blue 5's ring:

We're down to the final few cells of the puzzle, but we now have a dilemma. If we expand the remaining overlays and compare them to the remaining hexes, we actually have very little information to work with in solving the final rings and overlays.

What finally helped me here during the practice run was the "REMAINING" counter; we need only four blue hexes to solve the puzzle. If we examine the remaining hubs, we find that we still need two blue hexes for the Blue {5} at the bottom-center, and two for the Blue -2- ring at the end of the third row. So if we focus only on these two respective overlays, we can erase any hex not directly associated with these rings:

This now gives us enough information to complete the Blue 4 ring in the third row. The ring and overlay have now been left with only four active hexes, with three marked; mark the final one in this ring next.

We also get one more clue towards solving the Blue {5} at the bottom. The blue hex we just marked is also shared with this ring, changing the maximum reach of our chain of blue hexes. The orange hex directly to the right of the hub itself can now be eliminated:




The next clue we have is from the Blue 5 ring in the third row. If we expand the hub, it covers six active hexes. However, two of them border the same empty "1" cell along the ring's bottom-right edge and are the only orange hexes from which it can claim its blue hex. Since only one can be marked, the third orange hex is now a guaranteed blue hex.

This gives us a blue hex for the "1" at the top-right of the ring. We need to erase the hex on the top-right edge of the "1". Additionally, the -2- restriction for the next ring dictates that the cell below the one we just marked also has to be erased. That leaves the Blue 5 ring with only one final hex to mark:



We should be able to finish the puzzle now. We've finally given that Blue {5} a fifth consecutive blue hex, so erase the ring's final hex. Now, the "1" we revealed at the bottom-left of the Blue -2- ring has its lone blue hex. When we erase the orange hex on its bottom-right edge, the "0" guides us to the solution:



Hexes Earned for Completing This Puzzle: 20

We are now 2/3 of the way through Hexcells Plus and our 100% no-mistake walkthrough. At this point, we've learned all of the puzzle mechanics. The final two chapters simply combine them in ways that make us expand our logic beyond even what we have seen to this point. The final puzzles will really push us to the limit.




Two men holding a double-edged spear? From an overhead perspective? I'm not sure how to describe this one. We're given empty hexes pretty much at the points, with a decent array of line headers to manage.

We actually get several very easy moves to begin with. At the top, the -2- has our standard ring of three consecutive orange hexes, so mark the endpoints and erase the middle cell as usual. The "2" we reveal clearly shares the blue hexes we just gave the -2-, so we can clear three more cells from this. And while we can't solve it, -3- we just uncovered has only four consecutive active hexes around it, at least letting us mark the second endpoint.

On the far right side of the grid, we have a free "0" to work with; we know what to do with it. And finally, on the left side, the "2" has only two hexes it can claim, and it will also feed two blue cells to the "4" just over from it.





So at the top, there isn't much to tell us how to complete the -3- or the "2" we revealed below the first "2". We may want to try tackling the two Grid Overlays we opened on the left side. Both of these are governed by a "3". If we expand both, they each encompass the empty "4" cell and all of its active hexes. Remember: The hubs themselves are contributing two of its blue hexes. We can see, then, that the two additional hexes the "4" claims will complete the hubs' blue hex requirements. This means we can erase two more hexes, one from each overlay; erase the orange hexes within each not connected to the "4".

We get a "3" and a {3} from this. The dichotomy here is interesting; they each share two orange hexes with the "4". We can see immediately that the {3} will thus share at least one blue hex with it. However, if we mark both hexes shared between the {3} and "4", it becomes impossible to give the normal "3" the blue hexes it needs, because two of its own orange hexes would have to be erased. So to solve the {3}, clear the first hex in its chain and mark the other three:

So for the "3" at the top of this section, we can safely mark the two orange hexes it does not share with the "4". But we're still guessing as to which of its two remaining orange hexes will be marked.

What helps us here is that we've just given a third blue hex to a diagonal headed by a "3" extending from the top-center cluster of the grid. When we erase the rest of the line, we uncover a second "3" below the first, giving us only one choice for the first "3" to capture its third blue hex. It will be shared with the second "3", giving it a complete set and letting us erase the last cell on its bottom-right edge:

So what about that Blue 7 hub right above the {3}? Expanding it shows it covers six of the seven blue hexes it needs; the "4" on its upper-right edge and all of its active hexes; and one additional orange hex. So the fourth blue hex claimed by the "4" will, of course, also give this hub the last one it needs. As we have seen, we want to erase the lone orange hex not connected to the "4".

The "2" that this revealed is extremely helpful, since it pops up adjacent to the two blue hexes it needs. Now, erase the other four in its ring. This alone will let us solve the entire rest of this quadrant:





Now, just up and to the right of where we just were is an empty "1" cell with only one possible blue hex to claim; it's the one with a "?" on its upper-right edge. That blue hex will be shared with the "1" right next to the first, so we can erase the cell below it. Now, just solve the -2- that appears, then mark the last hex that the newly-revealed "3" needs. The Blue 3 that appears is problematic; leave it for now.

Solving that small cluster, however, does let us make some more inroads into completing the top section. See the diagonal headed by a "6" near the diagonal we solved a few minutes ago? Well, we have now reduced it to six active hexes. All that remains is to mark the last two it needs. Doing this will give the -3- and the "2" we abandoned earlier the last blue hexes they each need. Clear the orange hexes still adjacent to them. This reveals two additional empty "2" cells, each of which pop up with the two blue hexes they need. So clear the orange hexes from them, as well:

Our next step should be to complete that other diagonal from the right side of the top cluster, which is also governed by a "6". It, too, has been reduced to six active hexes; just mark the last one. Afterward, move to the right side of this section and give the "2" within the diagonal headed by a "3" the only two blue hexes it can claim.

We have no clear guidance for completing the top quadrant, and there's still too much uncertainty regarding that Blue 3 hub. So we'll mark the last obvious blue hex we have, which I alluded to in an image annotation earlier. Over on the right side, the "1" on top of the "0" we revealed in the next-to-last column has only one blue hex it can possibly claim. A Blue 1 hub. Hmm...


That Blue 1 overlay is our next focus. It falls into yet another diagonal governed by a "6", which contains seven total active hexes; three of them are marked. Four of them are orange, and two of those fall within the radius of the overlay. Therefore, it's impossible to complete the line's blue hex requirements without marking at least one inside the overlay. We can determine three things from this: 1) That the two orange hexes within the line outside the overlay have to be marked; 2) That whichever other hex we mark for the line will give the hub the sole blue hex it needs; and 3) That the overlay's four orange hexes not included within this line have to be erased. So here's what it looks like:

The "1" that we revealed on the bottom-left edge of the hub tells us how to complete the diagonal, and we can solve a huge chunk of this section by just using the relationships on the grid:






The Blue 3 hub at the bottom of this cluster is in an interesting position; when expanded, it covers only three active hexes, which happen to also border an empty "3" cell. So when we mark the last two the hub needs, we also give the empty "3" cell the last two blue hexes it needs. This alone will let us solve the entire cluster except for one last cell; to solve it, notice that it falls into a diagonal governed by a "3":

There's one last diagonal at the top that we can solve now, and it's governed by a "5". The line has the five blue hexes it needs; erase the last one. Sadly, we can't solve that empty "3" cell yet...

We have a trio of diagonals governed by a "4" from the right side now. The one on the bottom of this trio already has four blue hexes; when we erase the rest, we finally open up the bottom section a little bit:

We're almost at the end of Puzzle 5-1 and have just opened up the bottom section. Here's where we left off:



We'll work with the empty hexes we just revealed in a minute, but this last step also gives us enough information to complete the column headed by a "4" at the right edge of the top section. We've reduced it to only four active hexes, with one already marked; mark the other three.

Now, we just gave a third blue hex to the "3" near the top of the column. Erase the orange hex on its bottom-left edge; complete the "3" that this opens by marking the last hex in the cluster.

Move back to the bottom now. The "1" at the very bottom, as well as the "2" on the bottom-left edge of the "0" we revealed a minute ago, have only obvious hexes to claim. Can you get to this point with that information?



We ended up solving another of the diagonals from the right side with that sequence; we still have one more. If we mark it, we find that it has three blue hexes and three orange hexes. Obviously, we have to mark one of those final orange hexes.

There's a pair of empty "2" cells in the next line down from this one; the line we're trying to solve contains their only remaining possibilities for their second blue hex. The only choice, then, is to mark the center hex of the trio; we have to give the empty "2" cells a shared hex to complete them and still maintain the restrictions of the line header. With this, we can erase the next few cells using only the relationships on the grid:

Only a few hexes remain. We want to start working with the diagonals from the left side now. Start with the one governed by a "5"; it has been reduced to five total active hexes, meaning we need to mark the last one in the line. That's going to give a blue hex to a "1" we opened up a second ago. We can clear the hex above that "1" now.

When we do this, we can solve the next diagonal up, which is governed by a "6". The cell we just eliminated reduced it to six active hexes; so now, we again just mark the one that remains. That gives a second blue hex to the "2" we just uncovered. Now, erase the cell above the "2". The "3" we reveal will claim the section's final two hexes:


We're down to just a final cluster of seven hexes, five of which have to be marked. This is where those final Grid Overlays come into play. We want to start with the Blue 4 overlays first. Why? Because, as we'll see, they cover the same number of active hexes as blue hexes they need; two within each are already blue. This means we'll be marking two hexes per overlay:


And then there were three. We now need to expand the Blue 3 overlays; they each cover three blue hexes already, eliminating the two endpoints in that final line of three orange hexes. We'll mark the final cell to complete the puzzle.



This puzzle isn't extraordinarily difficult, but it is long. A lot of work can be done by just using the relationships between cells on the grid, but getting to those points typically requires referring to an overlay or a line header. Still, the logic used in this puzzle is not too complicated; it just requires understanding of all the mechanics and the ability to analyze the restrictions imposed on a certain cell or cluster before making a move.

Hexes Earned for Completing This Puzzle: 16

The next puzzle gives us a blanket that probably wouldn't be very useful in the cold weather...




Every single column of this puzzle is given a line restriction; additionally, we are given lots of free blue hexes. Unfortunately, every last one of them represents a Grid Overlay.

Careful examination at the column headers and the positions of the blue hexes shows us that we can erase a lot of cells from the get-go. We'll take the columns from left to right:

• The second column is headed by a "1" and contains a blue hex; erase the rest of the column.
  
• Moving closer to the middle are three columns each governed by a -2-; they each contain a blue hex in the middle. So we need to erase the hexes directly above and below the blue hexes in each column.
  
• The fourth column from the right is headed by a "4" and contains four blue hexes; erase the two which remain.

Which leaves us with this as our opening move:



It can be difficult now to determine just where to go next. There are so many hubs with overlays scattered around, and most of the columns and empty hexes by themselves give us incomplete information. However, within the middle -2- column of our trio, the two empty hexes that we just opened actually have their required blue hexes. So we can erase the remaining hexes from both the "1" and "2" we revealed within the column.

These eliminations leave the "2" near the bottom of the first -2- column in the trio with only two obvious blue hexes to claim; mark these next. Return now to the Blue 3 hub on the upper-right edge of the "1" we opened a minute ago. If we expand the overlay, it covers two blue hexes already; it also covers the "2" on its bottom-right edge and all of its active hexes. As we have often seen, the blue hex we mark to complete the "2" here will also complete the blue hex requirements for the hub. So erase all orange hexes within the overlay not connected with the "2":

We can eke out one more blue hex within this general section. The Blue 4 hub within the same -2- column we've been focusing on actually covers only four active hexes when expanded. We can go ahead and mark the last one within the overlay.

The Blue 3 overlay we just uncovered helpfully covers the three blue hexes it requires. So we can erase all orange hexes within its radius. When we do this, we can complete the Blue 4 hub on its bottom-left edge; when expanded, it will now cover only four active hexes. Mark the last one, but then notice that this particular blue hex falls into a column governed by a "3". We've just given it the third blue hex it needs; erase the rest of the line.

So now, in the first -2- column of our trio, the "2" about halfway down now has only one obvious blue hex to claim for its second. From here, expand the Blue 2 overlay three columns to the right, in a column headed by a "4". This overlay covers a slew of orange hexes, but it also covers one of its necessary blue hexes. Additionally, it covers the "2" on its bottom-right edge and all of its active hexes. The hub gives the "2" a blue hex; whichever hex we mark to give it the second will complete the hub's blue hex requirements. As usual, this means we're erasing all orange hexes not connected to the "2":

You probably saw that I've also cleared away the "0" uncovered during all of that; as a general practice, it's good to go ahead and do this right away, just to eliminate clutter.

Continuing on, then, the "1" near the top of the third -2- column has only one remaining hex to claim. We can't do much with the Blue 4 hub that this reveals, though.

We've done about all we can with the center portion of the puzzle for right now. Can we fill in the left side any further?

Let's start with the pair of Blue 2 hubs near the bottom of the first column. In actuality, we only need to look at the upper one; the lower one simply covers less cells vertically but covers the same ones horizontally. The overlay already covers one blue hex (the hub below it); it also covers the adjacent empty "3" cell and all of its active hexes. We can see that its third blue hex will also contribute a second blue hex to the hub. So again, all orange hexes within its radius not connected to the "3" have to be erased. Notice that doing so leaves the nearby "1" in the second column with only one possible blue hex to claim; mark it next.

Above the Blue 6 hub in the second column is a "3" with four possible choices for its blue hexes. However, the "2" in the fourth column restricts it further. The "2" already has a blue hex; the two orange hexes shared between it and the "3" are the only ones left for it to pick from. Since we know only one of those two hexes can be marked, the two hexes to the left of the "3", in the first column, become guaranteed blue hexes.



We can get one more blue hex in the first column. Expand the Blue "2" at the very top; it covers the adjacent "2" and all of its surrounding hexes, plus one additional orange hex further down the column. Since the hub gives the "2" one blue hex, only one of the other three surrounding it can be marked. That selection will give the hub its first blue hex; the second must come from the lone orange hex the overlay covers which is not attached to the "2". Mark this one next.

We still haven't done anything with the right portion of the grid. Maybe we can work our way back to the middle. Start with the Blue 2 hub at the top of the final column; this overlay already covers two blue hexes, so erase everything else within its radius. In the next-to-last column, we uncover a "1" two hexes down; since it borders a Blue 5 hub, we can clear the hex on its bottom-left edge.

As a final step for this sequence, both of the Blue 2 hubs in the bottom-right corner expand to cover the two blue hexes they need. So now, just erase every orange hex within their respective radii.






At this point, there isn't anything particularly obvious to work with. If you're working this puzzle mainly on your own and looking for a hint, the best advice I can give you at this point is to just expand different overlays and see if there is one you can do something with. The empty hexes and column headers still give us incomplete information to work with alone.

At this point, our most obvious move lies in the fifth column, which is governed by a "6"; the Blue 4 hub nearest the bottom covers its four necessary blue hexes when expanded, letting us clear the two orange ones in its radius.

It's not immediately obvious, but this opens up a partial solution to the Blue 4 at the bottom of the fourth column. This overlay now covers one blue hex, the two orange hexes from which the nearby "3" will claim its third blue one, and now only two additional orange hexes. We know that whichever hex we mark for the "3" will give the hub a second blue hex; so the third and fourth will now come from those two outliers. Mark them next.

We have now given the first -2- column in our favorite trio the second blue hex it needs; erase those which remain.

Puzzle 5-2 is taking us all over the place! We're just under halfway through it now:



Staying in the same general area, the Blue 7 hub about 2/3 of the way down the third column can now be solved. When expanded, the overlay now covers only seven total active hexes. Just mark the last three.

Finally!! We've given that "3" at the bottom the third blue hex it needs; get rid of that last orange hex on its bottom-right edge.

We can also complete the Blue 7 overlay just down and to the right of the one we just solved; it now also covers seven total active hexes. Just mark the final one.






At this point, we pretty much just have to keep looking at overlays and gradually weeding out the rest of the blue hexes. I'm going to list some easy ones to solve, with the necessary steps to do so:

• Blue 5, bottom of the fifth column (header: "6"): It covers five active hexes; erase the lone orange hex.
  
• Blue 6, middle of the fourth column (header: "3", already solved): It covers only six total active hexes and needs the final one to be marked.
  
• Blue 4, about 1/3 of the way down the seventh column (header: "5"): It covers only four active hexes and needs the final one to be marked.

There's finally enough information to work with some of the empty hexes and column headers. In the second column, the "3" about halfway down now has three blue hexes; clear the one on its upper-right edge. Also, in the sixth column, which is the first of our trio of -2- columns, the "1" two cells from the top has a blue hex, letting us erase the orange one on its upper-left edge.

Now, expand the Blue 4 overlay two cells down from that "?" we revealed; the overlay covers only four active hexes now. When we mark the fourth one, we also give the "2" on the upper-left edge of the overlay its second blue hex. In turn, when we erase the orange hex on the top-left edge of the "2", we leave the "3" near the top of the second column with only one obvious blue hex to claim. And when we mark that one? We give the first column, governed by an "8", the final blue hex it needs and can erase the two orange hexes which remain.

I'm also dimming out the hubs as we solve them; in some cases, several are solved at the same time. Let's expand the Blue 5 near the top of the first column; it now covers only five active hexes, so we can mark the last one. This will give the "2" at the top of the second column its second blue hex, so now, clear the one on its upper-right edge.

Let's now take the "3" at the top of our first -2- column. Yes, it is surrounded by four orange hexes, but the two on its left fall into a column governed by a "6". Since this column already has five blue hexes, we know only one of them can be marked. This also guarantees that both hexes to the right of the "3" will be marked. Marking them completes that whole column, as well.

One final move for this sequence. The Blue 5 overlay at the bottom of the first -2- column covers only five active hexes. Mark the fifth one. Doing this will give the second -2- column its second blue hex; erase the line's last orange hex at the top.





Let's now mark the two remaining hexes the "4" we just revealed needs. This will also give the associated column, which is also governed by a "4", the remaining blue hexes it needs. Erase the line's final hex at the bottom.

There are only two orange hexes on the left half of the puzzle. To solve them, it's easiest to use the Blue 3 overlay further down that column; it covers four active hexes, but three are already marked. When we erase the orange hex inside it, we have to mark the final hex to complete the adjacent "3".




The easiest move we have now is the Blue 3 overlay near the bottom-center; it covers three total active hexes, so we need to again mark the last one. Doing this gives our third -2- column a second blue hex, so now, we can eliminate the hex at the top of the column.

Again, it's hard to find another move from here. We can derive another blue hex from the Blue 4 overlay near the top-right. It covers two blue and three orange hexes; two of the orange hexes border a "2", which gets its first blue hex from the hub. Since only one of those hexes can be marked, we can safely mark the orange hex not adjacent to the "2".


The Blue 5 hub we just revealed is actually immensely useful, despite the large number of hexes the overlay covers. Expand it, and we can see that it already covers four blue hexes. Of the orange hexes it covers, two are the ones adjacent to that empty "2" cell we're trying to solve. As we've already established one of them has to be marked, we can erase all of the other orange hexes within the overlay. This now leaves the empty "3" cell near the top-center with only one obvious hex to claim for its third blue one; mark it now.

This next step is a bit of a stretch, but it does work if you think it through. Expand the Blue 4 overlay two cells below the Blue 5 hub we're working on. This overlay covers two blue hexes. It also covers the empty "2" on its bottom-right edge; all of its surrounding hexes; and the two orange hexes of another "2" three columns to the left.

It's easy to see that the third blue hex for this hub will come from whichever hex we mark for the "2" it borders. What is not quite as obvious is that the fourth one will come from whichever hex we mark for the "2" three columns over. These two orange hexes are the only ones from which that "2" can derive its second blue hex, dictating immediately that one of them has to be marked. As a result, we can erase the two hexes not connected to either "2":



We'll get one more elimination from the "2" that just sprang up; we already know that one of the hexes to its left has to be marked. The Blue 4 hub gives it one blue hex; so now, we can erase the hex below the hub.

Move now to the Blue 5 near the top of the final column; this overlay now covers only five total active hexes, giving us four more blue hexes to mark. Since one of them falls into a column governed by a "1", we can now erase two additional hexes:




Let's work the Blue 4 overlay below the Blue 5 that we just solved; this one already covers four blue hexes, so we can clear the two orange ones inside its radius. This also reduces the final column, governed by an "8", to only eight active hexes; mark the two which remain.

Additionally, the "3" near the bottom of the next-to-last column is left with only one final choice for its third blue hex. We want to mark it next. Now, expand the Blue 4 overlay on the upper-right edge of the "3". It covers only four active hexes; when we mark the fourth one, we give a second blue hex to the "2" positioned above the "3", and we can erase the orange hex from the top-left edge of the "2"--coincidentally solving that column in the process.

We're still not done! We're down to the final few hexes, but this puzzle isn't going down without a fight!



We now have enough information to solve that empty "2" at the top. Expand the Blue 4 overlay below it; it covers four blue hexes already, which lets us erase one of the orange hexes from the "2". Now, just mark the obvious blue hex we leave it with.

There's another Blue 4 hub three columns to the left, almost straight across from this one. When we expand it, it covers four total active hexes. Mark the fourth one, and we'll also give a second blue hex to those empty "2" cells further down. We'll erase the orange hex still shared between them.



Let's keep working our way down. The next Blue 4 hub about halfway down the same column also covers only four active hexes; again, mark the last one.

Let's move now to the Blue 7 hub in the third-to-last column; this one now covers only seven total active hexes, giving us two more to mark. These fall into one of the columns governed by a "5", giving it a complete set of blue hexes. So now, let's erase the column's final orange hex.




We can now solve the Blue 3 hub at the very bottom; expanding it shows that it covers three active hexes. When we mark the third, we also give the lone blue hex that the column, governed by a "1", needs and can erase the other two.

Only four orange hexes remain, with only one blue hex to mark. We can immediately tell that it will go to that pair empty "2" cells bordering the top two orange hexes. So we can erase those bottom two outright:





Two final hexes; which one gets marked, and which one gets erased? Several of the remaining overlays will give us the answer, but probably the simplest is to use either of those Blue 2 overlays at the bottom; they each cover two blue hexes already. When we erase the lone orange hex they cover, the final hex of the puzzle is marked.



This is definitely the hardest puzzle Hexcells Plus has thrown at us so far. I actually abandoned one solution as I was writing it because it was getting to be too convoluted; it turns out that, even in replaying it to write out this particular solution, that the puzzle simply forces you to readjust your strategy and approach it one small section at a time. There really isn't a "breakthrough" move that opens up large sections of the puzzle to solve; you have to gradually whittle away at it, constantly checking both overlays and line headers to see what is feasible to solve. It's both a marathon and difficult; if you don't undersand the puzzle mechanics by now, there's little way you will be able to work through this one without a guide of some sort. And the puzzles will only get harder as we enter Chapter 6 and work towards the end of the game.

Hexes Earned for Completing This Puzzle: 18

Whew. After the marathon that was Puzzle 5-2, we're down to the final 10 puzzles of Hexcells Plus. Think of Puzzle 5-3 as a prelude of what's to come.




We are given a total of four disconnected grids; many of the lines have restrictions to them; we get a decent number of empty hexes; and we also get a decent number of hubs.

We start with a couple of zeroes, so go ahead and clear them out first. We even uncover another one to help us out a bit more. Helpfully, we erase two cells from a column headed by a {4}, dividing the line up in such a way that we can now solve it. Only two active hexes remain above the cells we just erased; they clearly also have to be erased. This will leave four continuous active hexes further down which we now can mark.

In the left-center part of the main grid, where we have one of the first zeroes, there are a -2-, a {2}, and a normal "2" which can be easily solved. The -2- has only three continuous active hexes; the {2} has a blue hex at an endpoint; and the regular "2" only has two obvious hexes to claim:




Staying within the same general section, the "2" on the bottom-left edge of the Blue 3 we revealed a moment ago has only one obvious choice for a second blue hex. Also, the "1" on the bottom-right edge of the "2" we solved a second ago gains its blue hex from that "2", so we can clear the cell on its bottom-right edge. Similarly, the "2" we revealed on the bottom-right edge of the {2} has the two blue hexes it needs, letting us clear its lone remaining orange hex. When we do this, we uncover a -3-, with a standard ring of four active hexes; mark the second endpoint.

If we now expand the Blue 4 overlay right below the {2}, we find it covers the four blue hexes it needs and can erase all orange hexes within its radius:






We actually now have enough information to solve the third column of the puzzle, which is headed by a -2-. Within the fourth column, headed by a "5", the "2" about halfway down needs a second obvious blue hex; that gives the -2- column its first. Now, up in the narrow strip at the top-left of the board is a "1" which also needs an obvious blue hex; it will become the second for the column. Once these are both marked, we can erase the rest of the column.

The second column is also governed by a -2-. We've left it with only three orange hexes, and they are all consecutive. So we can now treat it the same way we would a -2- cell with only three continuous active hexes around it. The "2" that pops up in the middle of the column appears with its two blue hexes, and we can erase the final hex in this cluster.


Let's move to the Blue 3 hub we uncovered awhile ago, at the top of the section we've been working in. The overlay already covers two blue hexes; it also covers a "1" in the narrow strip above it and both of its active hexes. So we know that whichever hex we mark for the "1" will also give the hub a third blue hex. Erase the orange hex within the overlay not connected to the "1".

The "1" that this uncovers doesn't do anything as far as telling us which hex to mark up here; however, it does leave its associated column, itself headed by a "1", with only one active hex at the very bottom. We can at least mark it.





It isn't yet possible to determine which hex(es) will be marked to complete the pair of empty "1" cells at the top yet. We still haven't examined the Blue 6 below; let's expand it now.

This overlay covers two blue hexes already, as well as the -3- and all of its surrounding hexes. We can see that the -3-, then, will contribute one blue hex to the hub. The other three will come from the three remaining hexes not connected to the -3-; we can safely mark them next.

The blue hex at the top of this trio is connected to a "1", so naturally, we can erase the next cell up the line. The "1" this uncovers shares the same blue hex, which clears the two cells on its right. We uncover another "1"--which again shares that same blue hex--and another "0" to follow. Solve the -2- which appears on the upper-right of the "0" as before, then carry out the sequence for a few more cells:

From this top-central cluster now, we can solve two more diagonals. From the right are three lines governed by a "3". The lower two of these have enough information to fill them in. Of this pair, the top one has been reduced to only three active hexes; mark the two at the start of the line. The lower line already has three blue hexes, so clear those which remain.

This erases a hex from that -3- cell, leaving only one obvious hex to mark for it. The empty hexes we just revealed also establish some new relationships to work with. In the process of working through them, we'll solve this entire cluster and the highest of our three diagonals:




Let's see how much further we can progress from where we just ended. We have a pair of empty "1" cells at the bottom-right of the top cluster; the top "1" has only an obvious hex to claim, and it will be shared both with the second "1" and a -3-, the latter of which we'll be able to solve by erasing the hex below the second "1".

That's about as far as we can go using the relationships on the grid. However, we can complete another line; from the left side of the top cluster is a diagonal governed by a "2" which contains the two blue hexes it needs. Erase the three orange hexes at the end of the line; the "3" that pops up in the center of that trio only has three active hexes to claim, and that will let us fill out part of this section:

There are a few more things we can accomplish from that. First, if we expand the Blue 5 hub towards the left side of the right-central section, we find that it covers only five active hexes. Marking the fifth one gives a blue hex to a "1" at the upper-right edge of the highlight, so we can then erase the hex above the "1". Additionally, we need to give a second obvious hex to the "2" positioned two cells below that Blue 5.

Down the same diagonal line as the Blue 5 hub is a "1" with only one obvious hex to claim. When we mark it, we give its respective column, itself headed by a "1", the lone blue hex it needs. When we then erase the rest of the column, we leave only three active hexes around a {3} at the bottom. We can immediately mark those hexes, but to solve the "2" on the upper-right edge of the {3}, refer to the empty "1" cells above and below it:

One more diagonal to complete. At the top, there's a line governed by a "6" extending through the right side. The line now contains six blue hexes; erasing the final hex at the end leaves only one hex for the "2" on its bottom-left edge to claim.

The hex we just marked is the second blue one for this column, which is headed by a "3". The bottom two cells in the column border a "3" with four total active hexes around it. SInce it's impossible to give it three blue hexes without marking one of these two, we can erase the orange hexes within the column not connected to this "3".

We're about 2/3 of the way through this puzzle, looking for a move to break the rest of the puzzle open. Here's where we left off:



Now, the "2" near the top of this column has only two obvious blue hexes to claim. We also get one more diagonal to complete, also extending from the top section. This one is also headed by a "6" and has been reduced to six active hexes. Just mark the two at the end.

The column governed by "6" on this side of the grid has also been reduced to six total active hexes; this lets us mark five more. Doing this also gives the overlay for the Blue 6 hub within the same column a sixth blue hex, letting us erase the lone orange hex the overlay covers. This, in turn, reduces that hex's respective column, which is governed by a "4", to just four active hexes, letting us mark the two which remain there:

We can do a little more on top. First, the "2" on the bottom-left edge of the Blue 5 hub already has its two blue hexes, so we want to erase the hex on its top-left edge. Now, in the narrow strip at the top-right, we gave the "1" here a blue hex; when we clear the next hex over, we uncover another "1" with only one possible blue hex to claim. With this, the nearby Blue 5 hub now covers only five total active hexes when expanded, so mark the last one it needs.

The hex we just marked lies at the top of a column headed by a "2" and becomes the line's second blue hex; so now, erase the orange hexes at the bottom of it. The "3" we reveal at the very bottom is surrounded by only three active hexes; mark the two it still needs. That will give a second blue hex to the "2" above the "3", and we can erase its remaining orange hex.

The "2" that this opens itself has only one obvious second blue hex to claim. After we mark it, we can complete the column in which the "2" lies. It has been reduced to only five active hexes, so we now want to mark the hex at the top of the column.





At the bottom-right now, the "2" has its two blue hexes; the "4" we reveal by erasing the orange hex below it solves this section.

We can also solve the right-center section, as well. We have a trio of columns governed by a "3". The first one was just solved; the second now has three blue hexes, so we need to erase its last orange one. This leaves the "2" directly below the "3" uncovered by this erasure with an obvious blue hex to mark, also providing a third blue hex to the last column of our trio. When we erase that line's final hex, the "2" uncovered gives us the solution:

Believe it or not, that's about all we can do using the existing line headers and overlays. In order to proceed, we have to solve the column governed by a "5" near the left edge of the puzzle. It contains six total active hexes; three of the four orange hexes are adjacent to a pair of empty "2" cells at the bottom.

We need to mark three of the four orange hexes; the easiest way to figure out which ones is to run through the possibilities for solving the empty "2" cells. Each of them already share a blue hex to the left; what happens if we share the second one? Well, if we do this, we would have to erase two more cells, and the column itself would only be able to attain four blue hexes in total. So we have to mark the top and bottom hexes in this trio, then erase the middle one. The column's final hex on top will then be marked. And when we do this, we'll get enough information to solve the narrow strip at the top:

The columns governed by a "4" and a "6" on this side are particularly vexing, because we don't have enough information to finish that empty "3" cell between them. Each column needs two blue hexes apiece for a complete set. For either one, it is impossible to attain a complete set without giving at least one blue hex to the "3". However, we have to mark one per column; if we try to give the "3" two hexes from the same column, it becomes impossible to give the other column a complete set of blue hexes since two would then be eliminated. This means the bottom hex of both columns is now guaranteed to be marked.

We'll have to focus now on the center section to try and get any further. Start with the Blue 5 overlay in the center; it covers six active hexes, including four blue and two orange hexes. Its column is itself governed by a "6"; it contains seven active hexes, with four already blue. Since we know only one of the two orange hexes within the overlay can be marked, the orange hex at the bottom of the column is now a guaranteed blue hex.

Now, retract that overlay and expand the Blue 8 on its left. This one covers nine active hexes; four are blue, five are orange. Notice that it covers the same two orange hexes as the Blue 5 overlay did; we've already established that only one of those two hexes can be marked. That also guarantees the other three within this overlay have to be marked.


We've just given the -2- column here its first blue hex; we can immediately erase the cell below the blue one, which gives us a -2- cell. With only the standard ring of three consecutive active hexes, we can solve it like so many others. That reduces the column to two non-consecutive active hexes; marking the last one gives a third blue hex to the "3" above it, letting us erase two more cells.

This now lets us complete the diagonal governed by a "7" from the upper-right edge of the puzzle. We've left it with only seven total active hexes. Now, just mark the two which remain. As a final step, we can also complete that diagonal governed by a "5" from the top-left edge of the puzzle; it now has five blue hexes, so we can clear the final hex at the end.


We can now solve the Blue 9 hub near the bottom of the central column. Expanding it shows that it now covers nine total active hexes; we can now mark the final two it needs. This also gives the central column its final blue hex; erase the final cell from the column.

We still need to fill in the bottom. We do have a "3" with three blue hexes from our previous sequence; erasing the hex on its upper-left edge simply reveals another "3". What now?

This is where the Blue 8 up above comes in; when expanded, it covers six blue and three orange hexes. Two of the orange hexes belong to the "3" we just revealed, which already has two blue hexes. Since only one of these orange hexes can be marked, the third one the overlay covers is now a guaranteed blue. We also know the "3" must get its third blue hex from this pair, eliminating the cell on its bottom-left edge.

You may have been wondering as I was when we were going to solve that diagonal with an "11". That time is now; we just gave it the final blue hex it needed. Erase the final hex in that line. Marking the final blue hex the resulting "5" needs gives the adjacent "3" a third blue hex. Erasing its final orange hex uncovers a "4" which lets us easily solve these last two cells.


And finally, the end. Two of the three remaining orange hexes border a "1" that already has a blue hex. Erase them, then mark the final hex to complete the puzzle!



Hexes Earned for Completing This Puzzle: 20

This puzzle answers the question: "What do you do when the number of the Secondary Line Header is the same as that of the Primary Line Header?"


This time, we are given three identical grids. Every column has a header; and every single column is subdivided with Secondary and Tertiary Line Headers. We also get a single diagonal marked at the top-left corner of the second grid.



Now, before we start with the actual solution, I want to take some time to explain the line oddities introduced in this puzzle. In the image to come, I highlight three specific columns; understanding their intricacies is key to working with some of the lines we will encounter for the rest of the game.

Introduced in this puzzle is the concept of embedded lines whose header number is identical to that of the larger line. I want to start with the central column as perhaps the easiest introduction to this. The Primary Line Header is a "6". But so is the Secondary Line Header. The Tertiary Line Header, on the other hand, is a "2".

When working with divided lines, the headers for the embedded lines are actually more important, at least initially, than the Primary Line Header. The reason is that since they cover fewer cells, they have a larger impact on the lines as a whole. So let's apply this to the central column specifically:

• The Tertiary Line Header of "2" tells us that two of the column's six blue hexes must come from the bottom segment of four hexes.
  
• The Secondary Line Header of "6" tells us that, on the whole, six blue hexes must come from everything below it, meaning they can come from the middle or bottom segments--so long as we adhere to the idea that two come specifically from the bottom segment.
  
• The Primary Line Header of "6" tells us that, on the whole, the entire line must contain six blue hexes--again conforming to the requirements set forth by the embedded lines.

So in summary: The column as a whole will contain six blue hexes; two will come from below the Tertiary Line Header; six will come from below the Secondary Line Header, with four sandwiched in between them. What does that mean for the top section, then?

This actually means that zero hexes from the top segment of the column can be marked. In a situation where the Primary and Secondary Line Headers have the same number, nothing can be marked above the Secondary Line Header. In other words, the Secondary Line Header takes priority. We'll see a similar situation with Secondary and Tertiary Line Headers; when their respective numbers are identical, the Tertiary Line Header takes precedence over that segment of the line. With that introduction, then, here's a synopsis of the three columns I wish to highlight:



With that, there are a couple of ways we can actually begin the puzzle. It might be easiest to start with the knowledge we've gained in studying the columns. For example, if we start with the central column, we can open up a large section of the top grid.

We've already established that none of the column's hexes on top can be marked, and that the column's four hexes in the middle will be marked. So let's start there. Once we've done this, the "0" uncovered at the top-center forms the basis for a chain reaction. When we clear its hexes, we leave the "2" below it only two possible blue hexes. They are shared with the next "2" down the line, eliminating its extra two orange hexes. That, in turn, leaves the bottom "2" of this chain with only two possible hexes to claim. Marking those gives the pair of empty "2" cells we revealed on the left and right the blue hexes they need, erasing a total of four more. Finally, the pair of empty "2" cells we revealed on the bottom-left and bottom-right edges of the "0" will then have only obvious blue hexes to claim:

One thing I like to do is strike out the line headers for the segments already solved; feel free to do this as I am, or to leave them alone until the entire column is solved. Whichever way is easier for you to understand is the way you want to play it.

We want to do one last thing in the upper grid. Two columns to the left and two columns to the right of center, we have solved three out of four cells in the top line segments. The Primary Line Headers are "5" and "6", respectively. The way these columns are sub-divided, only one of their blue hexes can come from the top grid; they contain these required blue hexes. So now, erase the bottom hex in each of these line segments.

We're going to focus now on the bottom grid. It turns out that by solving some of the Tertiary Line Headers, we can almost complete the entire grid in a few steps! How?

Remember what we said: In lines that are sub-divided, the Secondary and Tertiary Line Headers are the most important, because they cover far fewer hexes than the Primary Line Headers. We can treat the segments covered by Tertiary Headers as small, individual columns on their own. Scan them; do any stand out?

Two of these segments are governed by a "3"; two are governed by a -2-; and one is governed by a -3-. The former two can be solved outright. As the segments governed by a "3" contain only three hexes apiece, all of their respective cells will be marked. Those headed by -2- can be treated just like a -2- hex with only three continuous active cells: By marking the endpoints and erasing the center hex. Finally, the -3- segment can be treated just like a -3- hex with four continuous active hexes, meaning we can mark the endpoints:

Believe it or not, we can now solve the entire bottom grid except for two hexes. Stay with me on this one; this is great. :-)

The next-to-last line segment in the grid is headed by a "2"; the one two columns to the left is our -3- segment; and in between them is an empty "4" cell with two blue hexes. The column requirements tell us that the "4" will capture one hex to its left, and one hex to its right. So within the next-to-last column, the very bottom hex is now a guaranteed blue; only one of its two hexes adjacent to the "4" can be marked if the -3- segment is to attain a third blue hex.

When we mark this cell, we uncover a Blue 7 hub. Expanding the overlay, we see that it covers five blue hexes. It also covers the two orange hexes to the right of the "4"--of which we already know that only one will be marked--and only one of the orange hexes to the left of the "4". Clearly, whichever one we mark to its right will become the sixth for the overlay; the seventh, therefore, has to be the one it covers within the -3- line segment. Since it will also be the third blue hex of the segment, its final hex will now be erased:

The seed has now been sewn; this grid is set up to be completely solved with the exception of those two orange hexes to the right of the empty "4" cell, which we won't be able to solve until later. The "3" that we just opened from completing the -3- line segment gains its three blue hexes from the column itself and from the "4". So here's what's going to happen: from this point until almost the left edge of the grid, we will reduce each respective line segment to only two active hexes. All of these remaining line segments are governed by a "2", meaning we'll be given only obvious hexes to mark within each line. This will remain the case until about the point we hit that empty "5" cell on the far left. Can you use that description to get this far?

We just made a sweep of the bottom grid in this puzzle, with only a few of its cells remaining:



The third line segment of the grid isn't much more complicated; the only trick is that we can only mark one of the two hexes adjacent to that "5", meaning the top hex is a guaranteed blue cell. That gives the "2" at the top of the next line segment a second blue hex, so when we clear the orange hex on the bottom-left edge of the "2", we have only one obvious hex to mark.

Let's now take the Blue 9 hub we uncovered at the top-center of the grid; the overlay covers nine total active hexes, so mark everything else within its radius:






We want to move to the middle grid now. The way the second and next-to-last columns of the puzzle are sub-divided, none of their blue hexes can come from it. So we can erase a total of six hexes just from this knowledge.

Let's start with the "0" we revealed in the bottom-left corner and erase its three hexes. Doing so leaves the "2" above the "0" with only two obvious blue hexes to mark, which will be shared by the "2" on top of this one. So clear the orange hex on the upper-right edge of the second "2".

Over on the right side, then, the "3" we just revealed at the very top of this line segment has only three active hexes to work with. Mark these next. In doing so, we'll complete the blue hex requirements for the line segment headed by a "4" to the left of the empty "3" cell. Erasing the bottom hex here leaves the "3" in the middle of the next column with only one final hex to claim. It will be shared with the "1"; erase the orange hex on its bottom-right.

We're given enough information now to completely solve the first and last columns. We've completed the Secondary and Tertiary line segments within each; the sub-divisions dictate that for each line, three blue hexes must come from the top grid. So now, mark the top three hexes of each column.



We've got a couple more line segments we can complete now, as well. On either side of the central column within the center grid are Secondary Line Headers of "3". They now have their combined three hexes from the center and bottom grids, meaning we can erase the final three hexes from each line. This completes their entire columns, too.

The "4" sandwiched in between the "3" and the second "4" on the right has only four active hexes to work with, so just mark the two it still needs. This also gives a fourth blue hex to the "4" below it, so erase the last orange hex in that line segment.

The "3" that this reveals pops up with its three blue hexes, so when we clear the two on its right, we get enough information to solve their respective line segment. We reveal another pair of empty "3" cells. The upper "3" has three blue hexes already, erasing the top hex of the line segment; the lower "3" needs to claim the bottom hex of this segment.



The Blue 6 hub we uncovered a minute ago only covers six active hexes when expanded; go ahead and mark the last one.

To the left of where we're working are two line segments governed by a "4". The one closest to the center already has three of the four hexes it needs; to get the fourth one, we have to give a shared hex to the "3" and "4" in the next column to the right. It's the only way to complete both the column and these particular empty hexes. After marking the second hex down, erase what's left.

We'll be able to solve the line segment to the left now. The "2" we just uncovered has only one choice for its second blue hex; the "3" pops up with its necessary blue hexes. This erases the two orange hexes on its left, and then the final hex of this line segment is marked.

One last step here: Expand the Blue 3 hub we just revealed. Since it covers only three active hexes, mark the last one.







We can now complete the bottom grid. See that lone diagonal governed by a "4" from the left side of the central grid? That line now has four blue hexes. When we erase the last one in the line, the final blue hex of the grid can be marked to complete that empty "4" cell.

We're almost done; all that's left is to fill in the top section. Notice how everything up here so far is symmetrical; in fact, we can work both columns governed by a "6" at the same time.

What these lines both have in common is that they now have five of their six necessary blue hexes. Their remaining orange cells each border a "3" and a "2" in an adjoining column, each of which also need a blue hex to complete them. So as we saw a minute ago, we have to share a blue hex between them, then erase the orange hexes left within each line:


Only a few hexes remain now. Even though the line headers are different, we can work the next two columns on each side in tandem; again, they each need only one blue hex apiece. In this case, we need to give a blue hex to each "3" we revealed at the top of the columns we completed, then erase the two remaining hexes within each:



We finally break the symmetry here, but the ending sequence is very easy. Over on the right, the "2" at the top of the third-to-last column has its two blue hexes, which erases the top two in the next-to-last column. Mark the final hex on this side of the puzzle.

And finally, to complete this level, the second column has three orange hexes adjacent to a "3" and a "2" in the third column. We have only one final hex to mark; again, sharing a blue hex between these cells is the correct answer. The final two are then erased to complete the puzzle.



On the whole, this puzzle isn't that difficult. The challenge lies primarily in being able to work with columns which are sub-divided into three line segments which each have a different header. The fact that every column of the puzzle is structured this way adds an additional layer of complexity. There are a couple of times when we have to use overlays to proceed, but none of these situations are particularly difficult. Once you understand how Secondary and Tertiary line segments feed into their respective lines as a whole, it isn't too difficult to work with them. The fact that so much of the bottom grid can be solved with relative ease after the opening steps are made also makes the puzzle more manageable.

Special Note: Some of you may have noticed some inconsistency in the screenshots as far as dimming the column headers. I realized I had accidentally forgotten to mark a hex in editing a particular screenshot and could not fix it without restarting the entire puzzle. In my attempt to reproduce everything up to that point, I missed dimming out those headers. I did not feel this to be a serious enough error to redo the puzzle a third time, but I wanted to mention this in the interest of full disclosure.

Hexes Earned for Completing This Puzzle: 20

Only two more puzzles to go in Chapter 5. At this point, the final chapter is unlocked if you would like to take a look at the hardest puzzles. Here's a look at our rather large next challenge.




We are given nothing to start this puzzle except column and line headers. Expect this one to be a relative marathon.

What really helps us to get the puzzle started, though, is that we are given two empty columns--that is, columns headed by a "0". Go ahead and erase all cells within both, and also from the "0" that is revealed in the shorter of the two columns. We're now going to cut the puzzle in half. One of the diagonals from the upper-right edge of the grid is headed by a "19". The line contains exactly 20 hexes, and we just erased one of them. So now, mark that entire line. As a final step for this sequence, also mark the fourth and fifth cells down within the final column. This one is governed by a {5} and contains eight total active hexes; if we were to plot out all of the combinations for five consecutive blue hexes, these two would be included in all of them.



All of that left us with a couple of possible avenues to explore. Over on the right, we gave a blue hex to a diagonal headed by a "1" and can now erase the rest of that line. The "2" we reveal at the end pops up right by the two blue hexes we just marked, letting us clear its two orange hexes. The "2" right next to it is then left with only two obvious blue hexes to claim, which will be shared by the next "2" up the line. To solve the pair of empty "1" cells at the start of the line, though, refer to the -3- uncovered from that last pair of erasures. We need to mark the second endpoint in its ring of four active hexes, which will give a shared blue hex to each "1":

We next want to solve the -2- we just revealed above the first "1" in that line. It has only a normal chain of three consecutive active hexes, so solve it like any such -2- cell. This gives a shared blue hex to the "1" on the bottom-right edge of the -2-; when we then erase its orange hex, the "2" on the bottom-right edge of the "1" is left with only one choice for its second blue hex.

Staying within the same section, the "3" above the -2- we just solved is embedded within a diagonal that is sub-divided into three separate lines. The overall line is governed by a "5"; the respective line headers dictate that three blue hexes will come from the end, one in the middle, and one at the beginning. We've already marked the blue hex it needs from the first line segment; clearing the two hexes at the start opens up another avenue.

The -2- that this uncovers has two active hexes along its top rim, and two along its bottom rim. One of those below it is already blue, so we can erase that adjacent orange hex. Doing this leaves the normal "2" we uncovered with only two possible blue hexes to claim; marking them gives the -2- its second blue hex, and we can erase its final orange hex. This uncovers a "4"; both it and the "3" below it will share the final orange hex in their respective clusters. From here, the "3" on the bottom-right edge of the -2- will share its obvious final blue hex with the "3" below it:

That's as far as we can go over here for now. Let's return to the longer of the two empty columns we cleared at the start. We have a trio of empty "2" cells, the middle of which got its two blue hexes when we solved the diagonal headed by the "19". So erase the two orange hexes on its other corners. We uncover another -3- on its bottom-right edge, again with a normal chain of four consecutive active hexes. Mark the other endpoint to give a shared blue hex to it, the third "2" in our trio, and the "1" below the third "2". Then, erase the hex on the lower-left edge of that bottom "2".

The "2" that this erasure reveals pops up with two blue hexes on its upper rim. So now, clear its two remaining orange hexes, as well as the one which remains bordering the "1" on its bottom-right edge. The nearby diagonal that is headed by a "2" is now reduced to two active hexes, so mark the second one, thereby also giving the "1" at the bottom of our empty column the blue hex it needs. Erase the orange hex on its bottom-left edge.

The "1" we just uncovered, as well as the "1" right above it, have only an obvious blue hex to mark. This gives the "3" above the hex we just marked its third blue, so erase the orange one on its bottom-left edge.





Let's now expand the Blue 6 hub above the -3-. The overlay covers the -3- and all of its surrounding hexes; the "2" on its upper-left edge and all of its surrounding hexes; four blue hexes; and several more orange hexes. The "2" and the -3- do not share any of their remaining orange hexes, so we know that one discrete hex has to be marked for each of them. These will complete the hub's blue hex requirements; so now, erase all orange hexes not connected to these empty hexes:

We uncovered a {3} with this, and it already has two blue hexes. We don't yet know which one will become the third, but we can certainly erase the orange hex on its bottom-right edge, since it isn't linked to the two blue ones. The "0" revealed tells us how to solve it. Now, we can carry out an extended sequence which solves a large chunk of the bottom section:


We can work back a little more towards the bottom-left corner now. In between our two empty columns is our embedded diagonal headed by a "3", which has been reduced to three active hexes. We still need to mark the one at the end. This gives a fourth blue hex to that "4" we opened up earlier. Clear the last orange hex from the "4"; the "3" that appears already owns three blue hexes, so erase the one on its bottom-left edge, as well.

Now, just right of the center column is one governed by a -2-. It turns out that our empty column of two cells is embedded within this large column. We gave the column a blue hex when we solved the diagonal headed by the "19" at the beginning. Erase the hexes above and below the blue one; the "3" revealed below the blue hex already owns three blue hexes, so we can clear its remaining three orange ones:

Our sub-divided diagonal now has only one active hex remaining, which we need to mark to give its second segment the blue hex it still needs. This move alone establishes several new relationships which let us work back towards the bottom. This is a lengthy sequence, so I'll split it up into a couple of screenshots:



We can continue our work down the grid for a few more hexes, but we'll eventually need to use the Blue 3 overlay at the bottom, and the Blue 4 overlay to the right, to continue:






So expand the Blue 3 overlay first. It covers two blue hexes, and we can easily see that the third will come from whichever hex we mark to complete the "2" below the hub. Erase the two orange hexes inside the overlay not connected to the "2". The "1" that we uncover already has a blue hex, which gets rid of three more; we can continue from here:

We're almost halfway finished; here's where we left off:



Here's where we need that Blue 4 overlay; when we expand it, we find that it only covers four active hexes. Marking the last one also gives a second blue hex to the "2" which stopped us at the end of the last sequence. This will let us work our way into the final column. Take heed when you get here, though; the "1" at the very bottom and the "2" right above it dictate how we complete its chain of five consecutive blue hexes. The "2" will have only two obvious blue hexes to claim, one of which will be shared with the "1". One final step to complete this half: the final orange hex lies in a diagonal governed by a "5", and it contains its five blue hexes. After solving the final two columns, erase that cell to complete this half of the puzzle.

Our next sequence involves a slew of erasures. First, the -2- column we referred to earlier now contains two blue hexes. Erase those which remain. Secondly, over along the top-left edge of the grid is a diagonal governed by a "5"; it's the one sandwiched in between another such line below it and a line headed by a "7" above it. This line now contains five blue hexes. We need to erase those remaining hexes, as well:

This kind of divides up the rest of the puzzle. Start with the "2" we just uncovered at the center of the grid; it pops up with the two blue hexes it requires. Erasing its two orange hexes lets us solve the embedded line within the central column. The existing relationships will let us extend further to the left and the right; it's a fairly extended sequence, so I will split this one up, too.


You know doubt see some steps we can make to continue that sequence; we're going to do that now. As we already know, we don't want to get too far ahead of ourselves in a complicated puzzle like this since that makes it easier to slip up. We'll carry the last sequence out as far as it will take us without having to use an overlay or a line header:



This solves a significant portion of the puzzle. But to continue, we'll have to use line headers again. Start with the column headed by a "4" to the right of the center; it contains four total active hexes. We need to mark the one at the top. This gives the "2" right below it a second blue hex, letting us erase two more. Both the "3" and the "2" which appear have only obvious final hexes to claim to complete their sets. We can continue down the right edge from here, too:

At the top of the -2- column we completed earlier are two empty "2" cells, the lower of which needs its second obvious blue hex. It, of course, will be shared with the top-most "2", giving it a second hex, as well. So now, we can erase the top hex from the center column. The "1" that appears will also share that blue hex; clear the cell on its bottom-left edge.

Now, if we go down the top-left edge of the grid, we'll come to two diagonals headed by a "7". The second one has exactly seven active hexes. Marking the final one at the beginning of the line gives the "2" directly under it a second blue hex. So when we erase the orange hex on its bottom-left edge, we leave only one possible hex for the "2" on its upper-left edge to claim--which coincidentally solves that blue hex's respective diagonal:

Let's finish the column headed by a "7" to the left of the center next. We've left it with seven active hexes, so mark the two at the top. That gives a third blue hex to the "3" in this corner; erase its last orange hex.

We're left with just five orange hexes at the top of the grid. A "3" borders two of them at the bottom; a "2" borders two of them at the top. Each of the columns are headed by a "4", and they each have three blue hexes. We have to complete the "3" from one column, and the "2" from the other. Which means that the orange hex not connected to either one is erased:


Okay, I'll say it: That "3" we just revealed is not helpful!! We're forced to try and determine at least one of the remaining hexes which cannot logically be marked. The "2" at the top and the "3" at the bottom of the cluster are probably the easiest places to try and do this; I'm going to use the "2" for this guide.

If we mark the hex directly below the "2", what happens? We would share a hex with the "3" we just revealed, but guess what? Both remaining hexes in that column would have to be erased! So there's our dud; erase the cell directly below the "2" at the top-center. We can now finish this section:



Down below, we still have a lone Blue 6 hub to complete. When we expand the overlay, it covers only six active hexes. Marking the last one gives a second blue hex to the "2" along the overlay's left edge. We can then erase the two hexes to the left of the "2". For now, though, this is a logical dead end.

So look just above this cluster, then. Don't try to solve the other Blue 6 overlay, though; there's still too much ambiguity. Instead, look to the diagonal headed by a "6", which is the next line down from the hub. Two of its three orange hexes border a "3" in that upper line, which has two blue hexes but no others from which to claim its third. The line itself already contains five blue hexes, so clearly, the sixth will also go to the empty "3" cell. Erase the first hex of the line.

Okay: This "3" isn't much more useful than the last one we uncovered. We now have a whole slew of restrictions to manage now. The Blue 6 overlay--which we now need to bring into play--has four blue hexes and four orange hexes. We have two diagonal lines headed by a "6"; one needs three blue hexes, the other needs one. We have two empty "3" cells, one of which needs two blue hexes, with the other needing one. And we even have an empty "2" cell which still needs a blue hex but which has two to choose from, one from each diagonal.

Go ahead and expand the overlay to see which four orange hexes it covers. How on Earth do we reconcile this? It comes down to the "3" that we revealed with that last step. It's impossible to complete its set of blue hexes without giving at least one to the lower line headed by a "6", but two of its active hexes come from this line. Do we mark both of them, or do we give it one from each line?

The overlay gives us the answer. We know that two of the orange hexes within its radius still have to be marked, but only one can come from the upper line; as it already has five blue hexes, it can only contain one more. Therefore, the other blue hex we mark must come from the lower line, also accounting for one of the three that this line still needs. The two hexes at the start of this line, then, are guaranteed to be marked, giving the empty "3" cell a complete set. Erasing its final orange hex sets up the solution:



We uncovered a "2" on the bottom-left edge of the "2" we just solved; it has two blue hexes, so erase the orange one on its bottom-left edge. The "1" that this uncovers is the key to several more moves that again do not rely on any overlays or line headers:

We are at the end of Puzzle 5-5, with only a few unsolved hexes remaining:



We have two more diagonals extending from the upper-right, and we can solve the one headed by a "5" now. This line has been reduced to five active hexes; we need to mark the three at the end. Doing this gives a third blue hex to a "3" located in front of the cells we just marked. So now, erase the hex on its upper-left edge. The "4" that this uncovers is surrounded by only four active hexes; mark the other two it needs. The Blue 6 hub that we open becomes the second blue hex for the "2" directly above it, letting us erase the two on its left edge:

Go ahead now and mark the second endpoint in the ring of hexes around the -3- we just revealed in the upper-left. Then, expand the Blue 9 overlay near the bottom; it covers nine blue hexes now. Erase the lone orange hex still within its borders, then solve that "5" which appears.




Only a few hexes remain unsolved now. Our next clue comes from the Blue 6 overlay near the top. It covers a "3" and all of its active hexes. The "3" needs only one more blue hex, but its only two possibilities lie within the overlay, which already has five blue hexes. So here again, erase the orange hexes inside the overlay not connected to the "3".


The "4" that appears here is surrounded by only four active hexes; marking the two it needs will also give two blue hexes to the "2" that we just uncovered, letting us erase the cell above it:

And now, the end. The column headers tell us how to complete the puzzle. The first column, headed by a -5-, contains its five blue hexes. Erase the top hex of the column, then mark the last one to finish this challenging level.



Hexes Earned for Completing This Puzzle: 22


And with that, we begin the final challenge of Chapter 5, which gives us a number of disconnected grids. The only free hexes we are given are concentrated around the middle; again, we have many line headers to work with.



This is where shared blue hexes can become a relative nightmare. The Blue 5 hub in the center dictates its entire respective cluster, which is teeming with empty "2" cells. We have no real choice but to start here, though; there simply isn't enough information anywhere else.

So start with the obvious: Our Blue 5 hub clearly gives the "1" right below it the lone blue hex it needs, so get rid of the hexes along its underside. Well, well, well: Three more empty "2" cells.

This might be enough to make your head hurt. Expand the overlay; it covers all of the empty "2" cells. The hub itself gives a blue hex to all of them within its overlay, meaning it is shared by six total empty hexes here. So we somehow have to mark five of the orange hexes without giving a "2" more than a second blue hex, and we have to be mindful of the fact that the ones we just revealed have candidates not covered by the overlay. So if we examine the hexes inside the overlay, one thing is clear: Most of the orange hexes are shared by a pair of the empty "2" cells. Only three of them are not.

In my practice run, I chose to focus on the orange hex located two cells straight up above the hub; in retrospect, it appears that this logic works for any of the three orange hexes only connected to a single "2". What happens if we mark it or erase it? If we mark this cell, the hexes on either side of it are erased; the empty hexes revealed would likely give us some guidance as to how the remaining cells inside the overlay are worked, and there is an avenue to give the hub five blue hexes.

But what if we erase it? Well, one of the hexes on either side of it would be marked; if we carry that out, we actually have to erase most of the hexes inside the overlay. At most, we would be able to mark only four blue hexes for the overlay. So eliminating this hex is not an option; we have to mark it, then erase the hexes on either side of it. We get another pair of empty "2" cells, but they actually tell us how to complete this cluster:



That was probably the hardest part of the puzzle! The Blue 3 we revealed at the top of the grid lets us solve the top grid. First, when we expand its overlay, it already covers three blue hexes. So the five orange hexes inside its radius now have to be erased. On the edges, we reveal two empty "1" cells; in from them are three empty "2" cells. The outer four cells of this chain have only obvious blue hexes to claim; the "2" in the center of all that will share two of them, and the top-center hex of the grid will be erased:

So returning to the center grid now, the empty "2" cells that were at the center-left and center-right of the Blue 5 overlay from before now have their two blue hexes, letting us erase a total of four more. Over on the left, we have a column headed by a "1" which contains only three total hexes, two of which border the "2" we just revealed. The "2" already has a blue hex, and the cells in that column are the only ones from which it can obtain its second. From this, the bottom hex of that column can now be erased.

We now open up a "0" to follow. When we erase its hexes, we can mark the top hex of the column, and since we now have a couple of blue hexes shared between different empty hexes, we can make a few more moves from this, too:





There isn't much we can do with the Blue 2 we just uncovered yet, but it does let us erase one more hex. The overlay covers one blue hex; the second will come from whatever we mark to give the "2" right above it a second blue hex. So erase the lone orange it covers that is not connected to that "2".

Let's now progress to the right of the Blue 5 overlay from the beginning. On the left, we uncovered a "2" and a "?" a moment ago; on the right, we uncover a pair of empty "2" cells. Their respective orange hexes fall into a column headed by--you guessed it--a "2". Again, the column contains only three total hexes; each "2" already has one blue hex. So as we've seen in previous puzzles, we want to erase the shared hex between them here. The top and bottom hexes are the ones to mark in this column; erase the center one. Then, go ahead and erase the hex below the lower "2" in our pair. The "2" that pops up here already owns two blue hexes, as well; if you're confident, go ahead and work ahead a bit here, but if you want to follow along with me, we'll hold off on moving on for just another moment.

We'll continue with the "2" that just popped up. As we said, it already has two blue hexes, so now, we'll erase the orange hex on its bottom-left edge. That "3" which appears has only two obvious blue hexes to claim; this almost gives us enough to solve the central portion of the grid, but we end up with a "3" that cannot be solved directly just yet:

We're already almost a third of the way done; here's where we left off:



We'll try to work towards the right side. The column headed by that {3} is a particularly interesting beast. We have an empty "3" cell to its left with two blue hexes already; its only two remaining orange hexes fall within this column. Naturally, only one of them can be marked, and it will become the endpoint in this column's chain of three blue hexes.

Using that, we can at least determine the maximum reach of that chain, and erase the hexes falling outside of it:







Since we erased a hex from one of the two possible chains of blue hexes we had available to us, we know how to complete the column. The hex on the upper-right edge of the "3" will be the endpoint, and we will mark the two hexes above it. The last two hexes of the column will then be erased:



This opens up some new relationships in both the central and bottom-right grids; we'll carry these through next. None of this requires either an overlay or a line header:






We'll be able to solve one of the diagonals extending from the lower-right grid now. Three of them are headed by a "5"; the bottom-most of these lines has now been reduced to five active hexes, so just mark those which remain in the bottom-center grid.

To progress the center grid now, we'll need to rely on a sub-divided diagonal. At the very top-right is a diagonal headed by a "3", which contains an embedded line headed by a "2". Well, with the exception of the very first hex in the line, every single hex between the Primary and Secondary Line Headers has been erased. So naturally, we'll mark that first hex in the line. This gives a blue hex to the "1" right next to it; clear the hex on its bottom-right edge now.

That still leaves us with some uncertainty as to how to solve the top-right section. We can use the Blue 5 hub to help us somewhat; the overlay covers four blue hexes already. We can see that the fifth has to come from whichever hex we mark for the "2" on the hub's bottom-right edge. Erase the two outliers within the overlay.

One of the empty hexes uncovered is a "3", and it leaves the "2" directly above it with only an obvious second blue hex to claim. When we mark it, revealing a Blue 6, it will be shared with multiple empty hexes. The "3" on its bottom-left edge gets its third blue one; erasing its final hex leaves the "2" on the bottom-right edge of the Blue 5 with only an obvious second hex to claim, as well. This ultimately gives us enough information to solve all but four hexes up here.

We've still done nothing with the top-left corner, but there is another diagonal, headed by a "4", which we can solve from this section. This line has been reduced to four active hexes, so we can mark those which remain.





So to complete this section, expand that Blue 6 overlay. It covers five blue hexes, meaning only one of the remaining orange hexes can be marked. It also covers two empty "3" cells and all of their surrounding hexes; each already has two blue hexes. Since each "3" needs only one blue hex, and we can only mark one hex, the only option is to mark the one hex shared between them. The other three will then be erased.

We've got another diagonal to complete, as well. The line headed by a "4" extending from the bottom of this cluster now contains four blue hexes; erase those which remain at the opposite end. Afterward, can you figure out how to complete that center section?




The Blue 9 overlay uncovered at the bottom-center of the main grid covers exactly nine active hexes, so mark the remaining five at the bottom . From here, return to the top-left of the center grid; the diagonal running along the bottom edge can now be solved. This one, governed by a "4", has also been reduced to four active hexes. This time, we need to mark those which remain.

After we do this, we give two blue hexes to the "2" near the front end of that line; we can erase the two orange hexes it still possesses. From here, find the "2" located on top of the Blue 2 hub; it now has two blue hexes, so erase the one on top of it.





Let's again start from the "2" on top of the Blue 2 hub; the empty "2" cells on its upper-left and upper-right edges have been left with only obvious blue hexes to claim for their second. So when we mark them, we give shared blue hexes to the "2" we uncovered at the very end of that last sequence. So now, erase the hex above that one.

The "4" we just revealed here pops up with four blue hexes already; erase its lone orange hex. Then, expand the Blue 9 overlay we just discovered. It covers six blue hexes already; we know that one will come from the last blue hex we give to the "4" we just revealed, so mark the two not connected to the "4" (you could also apply the same logic to the empty "3" cell on the bottom-left edge of the Blue 9). From here, just use the relationships on the grid to complete the cluster:

And now, the final hex in the bottom-right grid falls into a diagonal is governed by a "4", which now contains four blue hexes. Erase that last hex in the grid.

Only a couple of clusters remain, but the line headers don't completely solve them for us yet. We need to start with the dueling "2" and -2- in the bottom-left grid. We know that the hex under the -2- has to be marked. However, the two along its upper rim fall into a line headed by a "6", which contains four blue hexes and needs two more marked. Even though we're dealing with a "2" and a -2-, we can treat them similarly, since there are no other hexes for either one to claim. Since two of the three hexes have to be marked, we again will erase the center hex in the trio and mark the endpoints:

The final column of this grid, headed by a "4", now contains four blue hexes; erase the last one. The "1" which appears has a blue hex under it, so clear those to its left. Use the relationships to solve much of this grid:





We can now solve our embedded line, headed by a "2", and then use it to carry out the sequence further:







We solve the final two cells of this grid by using their respective diagonals, each headed by a "5". The top one has been reduced to five active hexes and needs the last one marked; the bottom one contains five blue hexes and needs its last one erased:




The final hexes will be solved using the remaining line headers. From the lower-left grid: Start by erasing the remaining hexes from the upper "6" line, and the lower "4" line; they each contain their required blue hexes. Then, mark all remaining hexes within the lower "6" line, since it contains only six active hexes:



See if you can finish the puzzle with this final image; I've labeled the only lines you need to worry about:



Hexes Earned for Completing This Puzzle: 24

Welcome to the sixth and final chapter of Hexcells Plus! Puzzle 5-6 is only really difficult at the very beginning; while the logic afterward is somewhat more complicated than earlier puzzles, the overall difficulty once you're past that opening sequence is not too high. Think of it as the calm before the storm. These final puzzles will be the ultimate test of what you have learned so far.


It almost looks like an infinity symbol, with some additional cells within each loop:



Almost every single line and column in the puzzle has a number governing its blue hexes. This is one of those puzzles that will require frequent rechecking of the line headers to move forward as we whittle down the hexes.

To start with, there are several moves we can make. Two of the inner columns are headed by a "2" and contain only two hexes, so we can mark these four to begin. We also, of course, want to erase the cells around the "0" given to us in the top cluster within the first loop of the puzzle. The "3" that is revealed above the "0" will have only three active hexes, so mark them all.

Now, from the left side of the puzzle is a diagonal headed by a "1" which extends across the lower-left section. Notice that two of the line's orange hexes border a "2" which has only three total active hexes surrounding it. The positioning means that it's impossible to complete the "2" without marking a blue hex within this line. So we can also erase all hexes within the line not connected with this "2".

Finally, to complete our opening, let's take the final column, which is headed by a {7}. The line contains nine total hexes, meaning we can mark several guaranteed blue hexes. If we were to number the cells from top to bottom, the following would be our possible combinations for seven continuous blue hexes:

• Cells 1 through 7
  
• Cells 2 through 8
  
• Cells 3 through 9

Since hexes 3 through 7 in this column have to be marked in all three combinations, we can now mark these five safely. So our opening looks like this:



You may have noticed that we just gave a blue hex at an endpoint to a diagonal headed by a {2} extending from the right edge of the puzzle. We know which hex to mark next for its second blue hex, and we can then erase the rest of the line.

Along the outer rim of the board at the bottom, we now have two empty "1" cells which each own a blue hex. Let's now erase the orange hexes which still border them. This reveals two more empty "1" cells; this time, we can mark the only possible hexes they can claim.

This will let us complete another diagonal from the left edge of the puzzle. The line closest to the bottom-left corner is headed by a "3". The line now contains only three active hexes after the one we just erased a moment ago. Mark all three.

One last step for this sequence. The three hexes we just marked gave blue hexes to each of three empty "1" cells. Erase the orange hexes still bordering them:






We'll deal with the "0" we just uncovered momentarily. For now, let's finish that small grid at the bottom of the first loop. All of those empty "2" cells have only obvious blue hexes to claim, meaning that all of this cluster's remaining unsolved hexes will now be marked. This also solves our diagonal headed by the "1" from earlier.

Now, the "0" is at the head of another diagonal governed by a "3", which only contains three active hexes. Mark the final one to complete the line. Now, erase the hex above the "0", then mark the cell above the "1" that this reveals:





Let's take care of the hubs we have revealed along the bottom rim. The Blue 2, when expanded, covers two blue hexes; erase the two orange cells that it covers. For the Blue 3 a little farther to the right, it opens to cover three total active hexes. Go ahead and mark all of them.

Returning now to the first ring of the puzzle, we can eliminate several more hexes from the columns headed by a -3- and a -2-. For the column governed by the -3-, we have marked two consecutive blue hexes. As the third cannot be connected to them, we can erase the hexes above and below that pair. Then, for the -2- column, we can see that its second blue hex will come from whichever one we mark for an empty "2" cell in the next column to the left, near the top of the puzzle. So erase the column's hexes which are not connected to that "2":

The diagonal headed by a "7", also extending from the left side, has now been reduced to seven active hexes. Mark its two remaining unsolved hexes now.

Now, move to the diagonal headed by a "3" right above that line. It already contains two blue hexes; the third will come from the empty "3" cell which gained its second blue hex from one of those we just marked. Erase the hexes within the line which are not connected to the "3".

This now lets us solve the column headed by a "5" from this same general section. Since it now contains only five active hexes, we can mark the last three:






We can build on some relationships established within the center cluster of the first loop. The "3" nearest the center was just given a third blue hex, so when we erase the orange hexes still bordering it, we leave only one obvious blue hex for both the "2" on its bottom-right edge, and the "3" at the left edge of the cluster, to claim. Completing the relationships solves the entire cluster except for three final hexes:

We'll be able to solve two more columns outright from this. First, our column headed by the -3- from earlier now has its third blue hex. Erase the last two from the line. Secondly, the column headed by a "4", located just to the right of the loop's center column, has been reduced to only four active hexes. We just need to mark the top hex of this column. In locating the correct column, be sure to start from the center column of this loop, and not the center column of the overall puzzle.

We can complete two more eliminations to fill out this sequence, then. The column headed by a "3", located to the right of the -3- column we just finished, contains two blue hexes now. Well, we can see that the third will come from a pair of orange hexes sandwiched between a "1" and a "2" near the top of the loop. The two hexes within the column which are not connected to them can now be erased:

We want to do some cleanup work now. Scattered around the board are several empty "1" cells that either need to have an obvious blue hex marked, or which already have a blue hex and need to be cleaned up appropriately. We can also go ahead and complete the -2- within the bottom cluster of the right loop:

We've completed a little over half of the puzzle now. Here's where we left off:



The last sequence solved one of the diagonals headed by a "3" from this side, and we can now solve the one a couple of lines above it, as well. This one just gained a complete set of three blue hexes, letting us erase the last two orange ones. The "1" that is uncovered owns a blue hex already; the "3" pops up with only three active hexes around it. Solve them accordingly.

Solving the "1" uncovers a "0", which lets us continue up its respective column. And solving the "3" lets us complete another diagonal. The two hexes we mark for the "3" give their respective diagonal, which is headed by a "6", the blue hexes it needs. We can now erase the rest of this line, as well. And in doing so, we leave the center column of the first loop, which is headed by a "4", with only four active hexes. Mark the last one at the top of the column:

We have enough information to solve the -4- on the right now; we'll come back here in a minute, though. Let's stay where we were and expand that Blue 5 hub at the top of the first loop. The overlay now covers five total blue hexes; erase the two orange hexes within its radius, leaving the two empty "2" cells at the bottom-left and bottom-right edges of the overlay with only obvious second blue hexes to claim. We'll be able to finish out this entire cluster from this.

From here, we can solve the column headed by a "3" at the right edge of this loop. We've reduced it to four active hexes, with the one at the bottom already marked. The three remaining orange hexes all border a pair of empty "2" cells to their left, which share a blue hex between them. This is easy; we need to mark two hexes here, so we do not want to mark the orange hex shared between them. Mark the top and bottom hex of this trio, then erase the middle one.

Let's now go back and solve that -4- within the second loop. We revealed another "3" on the opposite side of the "3" we uncovered a few steps back. This one is in a similar position, with only three active hexes. When we mark the third one, the -4- gets a third consecutive blue hex. So now, we need to erase the two orange hexes linked to the blue ones, then mark the final one in its ring. The empty "2" cells revealed share two blue hexes with the -4-. Erase the orange hexes above each "2", but stop at this new pair of empty "2" cells; we can't complete them yet...

However, the diagonal governed by a "7" can now be solved. We've reduced it to seven active hexes; mark the last two, which also gives a sixth blue hex to the final column. We still don't know which one will be the seventh, but we can now erase the bottom hex of the column, since it now falls beyond the chain's maximum reach:



I'm betting you already see how we're going to solve the final column. But for the record, the "2" that we just uncovered has only two obvious blue hexes to claim. One of them will become the final column's seventh blue hex; erase the last one at the top. The "2" revealed here will be solved just like the one at the bottom.

Now, from the upper-left section of the puzzle are two diagonals headed by a "7"; the one nearest the top-center of the first loop has seven active hexes. As before, mark the last ones in the line. As a last step for this sequence, we'll finish off the Blue 3 hub at the bottom of the second loop. When we expand it, it covers three blue hexes and only a single orange hex, which we will now erase.


This could have been done a long time ago, but we're going to solve the first column next. As you can easily tell, it contains only six active hexes, so mark the two pairs of orange hexes at the top and bottom. When we do this, we actually give our second diagonal headed by a "7" its seventh blue hex. So now, erase the final hex at the end of that line. This uncovers a "1", which already owns a blue hex; just erase the next hex over. And when we do this, the diagonal headed by a "3" from the upper-left edge is reduced to three active hexes, leaving only one left to mark to complete it, as well:

We can now solve the central column. Headed by a -3-, we just gave it a third blue hex; erase all of its remaining hexes. Once we get to this point, however, we'll have to use two of the overlays to continue.

At the center of the middle column is a Blue 4 hub; when expanded, the overlay covers three blue hexes, with the only two orange hexes falling into a column headed by a "5". This column contains three blue hexes already; the overlay establishes that only one of the two orange ones within its radius can be marked.

Now, at the top is a Blue 2 hub, whose overlay covers one blue hex, as well as the top two hexes of the column we're working on. The Blue 4 overlay actually dictates that this overlay's second blue hex will come from the top of the column we're working on. So the orange hexes covered by the Blue 2 overlay which fall outside of this column can now be erased:


The pair of empty "1" cells doesn't immediately seem helpful, but it gives us one important clue: One of them will get its blue hex in the column headed by a "5" to the left, the other from the column headed by a "4" to the right. The latter has been reduced to only five active hexes; the top two orange hexes border the empty "1" cells in question. The other conditons establish that only one of those hexes can be marked. This means that the bottom orange hex has to be marked; it so happens to complete one of the empty "2" cells we abandoned earlier. So if we carry this out, we get the following:

One other clue: The diagonal headed by a "4" from the upper-right contains four blue hexes, letting us erase three more. The "2" revealed within the column headed by the "5" tells us how to complete the column:





The diagonal headed by a "5" at the upper-right corner contains only five active hexes. After we mark the fifth, clean up the empty "1" cells in this section:






As you can probably already see, our final blue hex will go to the empty "1" cell at the top-right. Mark the only possible hex it can claim, then erase the final hex to complete the puzzle.



As an introduction to the final chapter, Puzzle 6-1 is not extremely difficult. It does require a lot of observation and some jumping around. In particular, it's important to constantly examine the line headers after working a sequence. As we saw frequently, it is sometimes possible to work on, or even solve, at least a couple of lines in quick succession.

Only five puzzles to go; the difficulty will really start cranking up from here.

Hexes Earned for Completing This Puzzle: 20

This is certainly one of the more minimalist puzzles of the game:



I remember that when I first worked through this puzzle, I marked all of the cells not attached to an empty hex without really having an idea of where to start. It turns out that this is a reasonable first step and that there is verifiable logic for it.

We're only allocated 16 total blue hexes. We can start by counting the number which will go to the empty hexes; since none of them are connected, there won't be any shared blue hexes between them. So we can add up their respective numbers: 5 + 4 + 2 + 1 = 12 total blue hexes which will be associated with the empty hexes. Well, it turns out that there are four orange hexes not attached to any of these; since we still need four more to come up with 16 blue ones for the puzzle, these are now guaranteed blue hexes:



Naturally, we reveal a series of overlay hubs. The Blue 2 overlay at the top-right covers only two total hexes, so we know to mark them right away. That gives us two towards our ring of five around that {5} cell. But there's no way yet to know which direction the ring will circle or which hex will be excluded.

Let's move down to the "2" in the third grid now. It lies in the center of two criss-crossing diagonals which are each governed by a "1". What that means is that the "2" will claim a blue hex from each diagonal; therefore, neither hex above or below the "2" can possibly be marked. Erase those next.

Let's make one more move for this sequence. The Blue 3 hub, when expanded, will govern three of the blue hexes required by the empty "4" cell in the second grid. So we know the fourth will come from the first column; the question is whether or not it will be shared between the two diagonals over this grid, which are again governed by a "1".

The easiest way to reconcile this is just to plot it out. If we were to mark the hex on the bottom-left edge of the "4", what happens? Well, since the other two orange hexes within these diagonals also fall within the Blue 3 overlay, this strategy fails. We would have to erase both of them, and then it would be impossible to give it three blue hexes as only two would be left. So the orange hex shared by these diagonals becomes our next elimination:

That last step also tells us that the cell on the top-left edge of the "4" has to be marked. After doing this, expand the other Blue 2 overlay between the first and second grids. It covers two cells in each grid, including the one we marked and two of the remaining orange hexes surrounding the {5}. The trick here is that it is impossible to complete the ring of blue hexes around the {5} without marking one from the overlay, which will also give it the second blue hex it needs. We don't know which one yet; what we do know, however, is that the hex on top of the "4" cannot be marked, and also that the two hexes around the {5} which are not covered by the overlay are now guaranteed blue hexes:

There's now enough information to solve the top two grids. We've reduced the central column, headed by a "3", to just three active hexes. We've also reduced the number of active hexes around the empty "4" cell to four. We need to mark all of their remaining unsolved hexes. That leaves one final hex in the top grid, which is now erased:



That leaves only three blue hexes left to allocate. Interestingly, we can now solve that empty "1" cell in the final grid. The first column of the puzzle, headed by a "4", has five total active hexes, two of which lie in the third grid by the empty "2" cell. However, the Blue 1 overlay to the right of that grid dictates that one of its blue hexes comes from the right, and the other from the left. The one we mark to its left will give the column only three blue hexes; we have to now mark the hex to the left of the empty "1" cell to come up with the fourth. Naturally, then, the hex to the right of the "1" will be erased:

So to get the final blue hex for the first column, expand the Blue 1 overlay we just uncovered; it covers only one active hex, so we need to mark it, then erase the final hex of the column:






We'll determine the final blue hex by now solving the criss-crossing diagonals influencing the empty "2" cell. We just gave the line extending from the top-right corner of that grid a blue hex. When we erase its remaining orange hex, the final hex of the puzzle is then marked.



This puzzle is tricky to get started, and it can be hard to reconcile all the various restrictions in play at any given time. It takes careful monitoring of both line headers and overlays to eliminate those hexes we cannot give to the few empty hexes we are given.

Hexes Earned for Completing This Puzzle: 20

Um, what...?




This puzzle is a different beast from any other we've worked so far. Even describing it is difficult; we get one large grid with several elongated extensions, and two very small grids to the lower-left and upper-right. But two other characteristics separate this puzzle from all others we have worked. First: Every single blue hex is the hub to an overlay. Second: Every empty hex we reveal is a "?". We get no numbered hexes whatsoever to work with. For these reasons, Puzzle 6-3 is one of the hardest in all of Hexcells.

We're given a lot of numbered lines for this puzzle. Because one of our main sources of information--numbered empty hexes--is effectively eliminated, mastery of overlays is critical. We'll need to use them in conjunction with the line headers for the duration.

To begin, we want to scan all of the line headers carefully, and see which ones we can at least partially, if not entirely, solve from the outset. There actually are several lines which can be completed right away. Three are diagonals extending from the left side; one is a column not far to the right of the center.

The column in question is headed by a "0"--an empty column--which lets us erase five hexes. Go ahead and do this now.

Now, extending from the bottom-left section are three diagonals, which are headed by, from bottom to top, a "1", a "3", and a {10}. The line headed by the "1" already has a blue hex--a Blue 2 hub--so we can erase its two orange hexes. The line headed by the "3" has only three active hexes; mark the two orange ones within it. Finally, the line headed by the {10} actually contains exactly 10 hexes, so we need to mark all of them. The notation seems superfluous here; if the line contains the exact number of hexes as blue ones it needs, it stands to reason that they'll all be consecutive...

Don't be daunted by the number of hubs we've opened up; we still don't need to focus on them immediately. There are still several more lines that we can at least partially solve. We'll work on as many lines as possible to whittle down the hexes before focusing on the overlays.

Two of the lines are sub-divided into two segments. The first is the column headed by a "5", which forms the very first column we see in the upper part of the puzzle. The Secondary Line Header is a "3", meaning that the top segment has to get two blue hexes. Luckily, there are only two blue hexes up here; mark both.

The second sub-divided line is the diagonal headed by a "7" at the upper-right corner. The Secondary Line Header is "3", so we need four blue hexes above it. There are only four active hexes between the Primary and Secondary Line Headers; again, mark them all.

Now, right around the center of the puzzle is a column headed by a {2}, which contains a blue hex near its center. As only two possible hexes can be linked to the blue one, we can erase the rest of this column.

As a last step for this sequence, we'll take the diagonal headed by a {3} at the right-center. This line contains only four total hexes; since that leaves only two possible combinations to give it three consecutive blue hexes, we at least know that the two middle ones are guaranteed to be marked.



Over on the right, we have a column headed by a "1" to which we have just given a blue hex. Erase the rest of that column.

Okay: We've officially come to the point that we cannot solve any more lines. We've still made a significant dent in the puzzle. But now, we have no choice but to start working with the overlays. If you're working through on your own, don't forget to recheck the line headers occasionally to see if any more can be completed.

Let's start with the Blue 4 overlay to the left of the column we just completed. When we expand it, it covers exactly four active hexes, so make sure they are all marked. After doing this, we will be able to solve another diagonal. Above the diagonal headed by the {3} is one headed by a "2"; we just gave it a second blue hex. So now, erase the rest of the line:


Now, expand the Blue 5 overlay on the bottom-right edge of the Blue 4 we just solved. This one covers exactly five blue hexes; marking the other two also contributes a blue hex to the other column governed by a "1" in this section. Erase that column's final hex.




We can actually use either the Blue 4 hub off by itself or the other Blue 5 hub to figure out this next sequence. Expanding either one shows that it covers its required number of blue hexes; when we erase the only orange hex covered by it, we can then mark the final cell in our diagonal headed by the {3}. Then, if we expand the Blue 6 overlay, we find it covers six active hexes, allowing us to mark the only orange hex it covers.

Before continuing, I want to advise you to dim out the overlays as you complete them. As we are already seeing to a degree, completing one is often enough to complete at least one other. Dimming them quickly will prevent possible confusion between overlays that are completed, and nearby, overlapping overlays that may still need a blue hex or two.

One last overlay we can complete over here is the Blue 3 a little farther up. This one also covers only the same number of active hexes as blue hexes it needs; mark the last one.

A little further to the left is another Blue 5. We can't completely solve it just yet, but we can get some information out of it. Let's expand it; it covers six active hexes, three of which are blue. Two of the orange hexes it covers fall into the diagonal headed by a "2" here, which already contains one blue hex. So we can't complete the hub's full set of blue hexes without marking one within this diagonal. Since it will become the line's second hex, we can erase the orange hexes within this line that are not connected to the overlay; we can also mark the one hex inside the overlay that is not included within this line:

You probably saw in that screenshot that I've also expanded a Blue 3 overlay farther down. There's a reason for this. We just marked a third hex in the column headed by a "4" here. This column runs through that Blue 3 hub. When expanded, that overlay covers four total active hexes; two are in the same column, and two are in the column to the left. So naturally, we cannot complete its blue hex requirements without giving its respective column the fourth blue hex it needs.

This sets up a few different moves. First, both hexes covered by this overlay in the column to the left of the hub have to be marked. Secondly, all hexes within the column headed by the "4" which are not connected to this overlay have to be erased. And finally, when we do this, we leave the diagonal headed by the "2" with only one final hex to mark, solving both it and the Blue 5 overlay above:

Now, the two hexes that we marked for that Blue 3 overlay each fall into diagonals headed by a "2", and they also happened to provide the second blue hexes for each. So we can erase the rest of the hexes within each line. We can then complete that Blue 3 overlay by just marking the last active hex it covers. This consequently also solves the column headed by the "4":

The last sequence puts us at about 1/3 completion; here's where we left off:



We can use either Blue 2 overlay at the bottom to solve that last orange hex. Since each covers two blue hexes, we can erase that one immediately.

Now, expand the Blue 2 hub a little farther up. This one covers three active hexes, including a blue one. But what's important is that the two orange hexes fall into a diagonal headed by a "5", which already contains three blue hexes; the overlay tells us that only one of the orange hexes in the middle can be marked. So the one at the end of the line is now a guaranteed blue hex. When we mark it, we give a third blue hex to its respective column, which is headed by a "3"; we can now erase the two hexes at the top of the column:

Let's briefly shift focus to the small cluster at the top-right corner. We only have three orange hexes left to solve; the trick is in determining which overlays to use.

If we expand the Blue 2 overlay on the right edge, it covers a blue hex and two of the three orange hexes. This is important; if we expand the Blue 5 on the bottom-right edge of the "?", it covers all three orange hexes but also three blue hexes. We've already established that only one of the two orange hexes shared between the overlays can be marked; the one directly above the Blue 5 hub is thus a guaranteed blue hex.

The cluster's final blue hex will now be determined by the Blue 4 right under the Blue 5. This overlay already covers four blue hexes, plus one orange hex. Erase that orange hex, then mark the final one in the section:





We still have one remaining Blue 4 hub trailing off of that cluster which we can't immediately solve, but it does give us some new information. The two orange hexes this overlay covers falls into a diagonal headed by a -5-, which already contains four blue hexes (One is separated from the rest; even if we end up with four blue hexes together, it still meets the requirements. This is why we can't immediately determine which one to mark.). We can at least erase the line's two remaining orange hexes outside of the overlay.

We'll now move to the Blue 5 hub near the end of our {10} line from the beginning. It expands to cover five blue hexes and one orange hex. Erase this one next.






We can't really solve any of the remaining overlays in the center part of the puzzle yet; they each cover far too many hexes to reliably eliminate at this point. There also isn't enough information for us to complete any more lines. So we'll need to start on the left side and work our way back over.

Let's just start with the Blue 2 hub at the start of our {10} diagonal. It expands to cover two blue hexes, letting us erase two more. The Blue 3 next to it then expands to cover the three blue hexes it needs; erase the lone orange hex within its radius, too.




We now leave the column headed by a "3" over here with just three active hexes; mark the other two. Afterward, expand the Blue 5 overlay at the top of the column; it will now contain five blue hexes. Erase the lone orange hex it covers.





So now, let's take the next Blue 5 hub down the {10} diagonal; this one contains only five active hexes, so mark the last one. Do the same thing with the Blue 7 hub next to the Blue 5:






The Blue 3 hub we revealed just above this line is extremely helpful now. If we expand it, it covers three blue hexes and five orange hexes, which we now get to erase.

Next, expand the Blue 6 hub a few spaces down in the next column to the right. This overlay covers five blue hexes already. Of the four orange hexes it also covers, two fall into the diagonal headed by a "4". This line contains three blue hexes already; the two orange ones are the only hexes left to solve. So one of them will also give the Blue 6 hub the last blue hex it needs. We can erase the other two orange hexes within the overlay now:

Back on the far left, we have enough information to solve the Blue 1 overlay; we've left it with only one active hex, which we will now mark. Subsequently, the Blue 2 near the bottom of the next column will expand to cover two blue hexes, letting us erase three more:




The Blue 4 hub just over to the right of the Blue 2 we just worked now expands to cover four active hexes, so mark the fourth one.

At this point, it takes some degree of experimentation with the remaining hubs in this section to determine which one gives us a move we can make with 100% certainty. The one we want is the Blue 4 hub positioned at the bottom of the column headed by a "2" near the center. It covers four total active hexes. When we mark the last one, we give our diagonal headed by the "4" the last blue hex it needs and can erase the last cell in that line:

It turns out that one of the columns governed by a "3" in this section also gained its third hex from that last sequence, so we can now eliminate the top two hexes of the column. Additionally, at the bottom, we can expand any of those overlays and find that the final orange hex down there has to be marked. That orange hex in the bottom-left corner, however? We won't be able to solve it until we're in a position to complete its associated diagonal.

This leaves a bit of a dilemma as to where to go now. If we expand the remaining Blue 4 overlay on this side, it has three blue hexes and two orange ones. Those two orange hexes fall into a column headed by a "2", which already contains one blue hex. Since we can only mark one of the two inside the overlay, we can at least erase the top hex of the column.

Let's just go up to the top-left corner now. The Blue 2 overlay expands to cover one blue and two orange hexes; naturally, only one of those can be marked. If we expand the Blue 3 below it, though, it covers those same two orange hexes, plus one more. Since we can only mark one of the two shared between them, that extra cell is now guaranteed to be marked:


There's no real way to know where to go next; again, it takes some experimentation to determine what will give us a logical move we can make, and even then, it's far from obvious.

Just below the center, in the column headed by a "5", is a Blue 3 hub; in the next column to the right just about right in the center is another Blue 3 hub. If we expand the lower Blue 3 overlay, it covers two blue and four orange hexes; take note of the four orange hexes, then retract it.

Now, expand the upper overlay. This one also covers two blue hexes, but it also covers seven orange hexes--including the four covered by the lower overlay. This means whichever hex we mark for the lower Blue 3 will also give the upper Blue 3 a third blue hex. So we can erase the orange hexes that are not shared by these overlays. And when we do so, we leave our column headed by the {2} only one final hex to mark:

We're nearing the end of this puzzle, but it's not going down without a fight!



Let's now expand the Blue 5 overlay within the column headed by the {2} that we just solved. It covers six active hexes: Two are blue, four are orange. Two of the orange hexes fall into the column headed by a normal "2" to the left; this column contains one blue hex already. Since only one of those hexes can be marked, we can safely mark the other two orange hexes inside the overlay's radius.

Now, expand the Blue 3 we just uncovered at the upper-right corner of this overlay; this one covers three blue hexes, so we can erase the lone orange hex inside its radius:






That last sequence helps us by reducing the number of active hexes covered by the Blue 3 overlay just up and to the right of the one we just solved to three. When we mark the other two, we also give a fifth blue hex to the column headed by a "5" and can erase its final hex:




Just over to the right, the column headed by a "3" has now been reduced to three active hexes. Mark the last one. Now, expand the Blue 2 overlay at the top of the column; it covers only two active hexes, so again, mark the last one it needs.

After this, move down a little bit, to the diagonal headed by a "3". We've finally given it three blue hexes and can erase the final hex from the line. When we do this, we also leave the diagonal headed by the -5- with exactly five active hexes and can mark its final hex, as well.

One last move for this sequence. On the right side, we still have a diagonal headed by a normal "5". The line contains five blue hexes now. Erase its final orange hex.



We've now filled in a lot of information in the center of the puzzle. It's worth rechecking the overlays to see if we can complete any more of them. Let's now expand the Blue 2 near the top of the {2} column; it now covers two blue hexes, so we can erase the orange hex it covers. This one falls into the column headed by a normal "2" to the left, which we have just reduced to two active hexes. Simply mark the second.

Now, expand the Blue 3 overlay pretty much dead center; it covers three blue hexes now, letting us erase two more. Erasing them reduces the diagonal headed by a "5" from the top-left corner to exactly five active hexes. We now can mark the last one in that row. We can also erase the final hex in that corner; we had already established long ago that we could only mark one more hex in this section.



Only a few orange hexes remain, but we still have several lines and overlays to complete. If we now expand the Blue 3 overlay located about halfway down the diagonal headed by the -5-, it covers its three blue hexes. Erase the orange hex it covers. Doing so leaves the next line up, which is headed by a "5", with five active hexes; mark the last one in that line.

Marking that hex also gave the column headed by a "2" its second blue hex; erase the final hex in that line. And when we do this, the diagonal headed by a "7" is reduced to seven active hexes. Now, we can finally mark that final hex in the bottom-left corner!




A Blue 0 hub?? Why couldn't that have come up a long time ago? And why aren't there more of them? :-P

In any event, we're finally done with this marathon of a puzzle. To determine the puzzle's final blue hex, we just need to realize that the diagonal headed by a "5" now contains five blue hexes. When we erase the final orange hex from that line, the puzzle's last hex can then be marked.



As stated earlier, the fact that there are absolutely no numbered empty hexes helps to make this a monster of a puzzle to work through. Add to that the fact that every single blue hex opens up into an overlay, and you have an incredibly challenging setup. Let's be honest: If you can't reconcile the overlays with the line headers to work through the hexes, this puzzle will make short work of you. And we haven't even hit the hardest one yet. Only one more puzzle stands in our way before we reach the ultimate test of Hexcells Plus.

Hexes Earned for Completing This Puzzle: 22

We're down to the final three puzzles of Hexcells Plus. This is the last one before we reach the game's ultimate challenge.


This one gives us four separate grids; it's almost like a giant sphere with a cross-cut in the middle, and a Blue 2 hub dead center:



In my practice run before attempting to write this out, I found that the hardest part is keeping up with the relationships on the grid, as well as a few tricky setups concerning overlays. The relationships among the cells are more complicated than in earlier puzzles, and it's harder to miss a shared hex. We'll go through the harder ones in detail.

There are enough line headers that it's a good idea to scan them and see if any can be at least partially worked from the outset. Several of the lines are also sub-divided. Interestingly, we also have a few very short line segments which have headers, while their other segments are not headed by anything.

There are actually four separate line segments we can solve right away. Three of those are in the bottom half of the grid. Of those, two are empty columns (headed by a "0"). The third is headed by a -2-; as it only contains three total active hexes, we know to mark the endpoints and clear the middle hex.

The final line we'll work in our opening extends from the upper-right quadrant of the puzzle and is headed by a "6". This particular line is also sub-divided; notice that both the Primary and Secondary Line Headers are "6". In a situation where both headers are the same number, all of that line's blue hexes will come from below the Secondary Line Header. So we can now erase all hexes in between the line headers:

You probably saw that I have also cleared away the "0" that is uncovered in the lower-left; in general, you should go ahead and do so whenever a "0" appears just because it gives you free hexes to erase, whittling down the puzzle with no effort.

We can start working on that chain of empty hexes we revealed in the upper-right quadrant now. Start with the -2- right on that upper-right edge; it's positioned to only have three active hexes around it. Along its bottom rim are two, but the third is on its top-left edge. Since the two blue hexes have to be separated, we can mark this one immediately. It will, of course, be shared by the second -2-, as well. Now, since the second -2- has two discrete pairs of active hexes, marking this one allows us to erase the second hex in its upper pair.

Next, erase the three orange hexes from the "1" that just appeared, then mark both hexes along the bottom rim of the {2}. The second -2- gains its second blue hex from this; we need to erase its final orange hex, leaving only one left for the -2- on the edge to claim. Carry out the relationships in this grid for a few more moves to come up with this:


We can actually continue building upon that last sequence, and when we do, we'll be able to partially solve one of the columns up here. The column headed by a "3" in the middle of the grid is sub-divided; the Secondary Line Header is a "1", meaning that two of the blue hexes have to come from this section. Within this line is a pair of empty "2" cells, the bottom of which has only two possible blue hexes to claim. When we mark both, this line segment gets the second blue hex it needs, letting us erase the rest of the column between the two line headers. Both hexes we just marked are also shared with two empty "2" cells apiece; this lets us erase more hexes and progress this section further:

This solves most of the grid. We can try to fill out the bottom-right corner by starting with the "1" on the upper-right edge of the "0" at the bottom-center; it has only one obvious hex to claim. Two spaces above the cell we just marked is a "3" with only three surrounding active hexes, so now, we need to mark the third. Both blue hexes we just marked are shared by the "2" sandwiched in between everything, so now, we can erase its last orange hex.

We get another "2" from this elimination, and it shares those exact same two blue hexes! So we erase the two hexes to its right. Now, the "3" two spaces up from this "2" has only an obvious third blue hex to claim; we can finish this cluster now:





Now, back at the top of the column headed by a "5" in which we were working earlier, we have now left the "1", "3", and "4" with only the same number of active hexes as blue hexes they need. The end result is that we get to mark all three top hexes of the next column to the left, which is also headed by a "5". At the bottom of the column is a -3- with only four continuous active hexes around it; one endpoint is already marked, so now, mark the second.

We can now use the Blue 4 that we uncovered earlier to give us our next clue. If we expand it, the overlay covers that -3- and all of its surrounding hexes, as well as three total blue hexes. The fourth will naturally come from the one we use to solve the -3-, so the two orange hexes that it covers which are not connected to the -3- can be erased:

The -3- that we uncover about halfway up this segment pops up with two consecutive blue hexes along its top rim; obviously, that third active hex on its top-left corner cannot be marked. Erase this one, then mark the one under it. That will give the lower -3- its third hex. Go ahead and solve the "4" that appears by marking the rest of its four active hexes, then work up the left side of the grid:

Two things now. First, our column headed by a "5" which contains the two -3- cells is sub-divided; the Secondary Line Header is a "1", meaning that four blue hexes have to come from the upper line segment. We have now done this; erase the two orange hexes within this line in the bottom-right quadrant just before the Secondary Line Header.

Secondly, expand the Blue 2 hub in the very center of the board. The overlay now covers the two blue hexes it needs; erase all orange hexes that it covers, and clear away the zeroes, as well:






The "3" that we opened up right above the Secondary Line Header of this column has only three active hexes bordering it. Marking them sets up a series of eliminations as we uncover several empty hexes which share these blue ones. We can't really use the hubs we've just revealed yet, though. So our last move for this sequence will be to mark the three obvious blue hexes that the "3" at the very bottom of the third-to-last column will claim:

We're about 1/4 of the way through now:



There are a couple of places we can go now; we can move back up to the top-right quadrant, or we can move into the lower-left quadrant. For this walkthrough, we'll go back up to the top and expand the Blue 6 overlay we abandoned earlier.

This is an interesting case. The overlay covers three of its six blue hexes already. It also covers six orange hexes. We can easily see that the fourth blue one will come from whichever hex we mark to complete the empty "4" cell on the hub's upper-right edge. Since there are two possibilities here, that accounts for two of the six orange hexes.

Three more of the orange hexes fall into the final column of the top-left grid, which is headed by a "4". This line is sub-divided such that this end of the line will only get one of the four blue hexes; the Secondary Line Header of "3" is positioned such that the others have to come from the bottom-left grid. Now, the fact that, outside of these, there is only one more orange hex to account for within the overlay means that one of these three has to be marked. We can erase the two hexes in this segment that are not included within the overlay. We can also mark the sixth orange hex that it does cover:

So within the column headed by the "4", we have now left the "1" at the bottom of this segment with only one choice for a blue hex. The "1" on its bottom-left edge will share this blue hex, as will the "2" above it. When we erase the orange hex from the upper-left edge of the second "1", we uncover a -2-, which tells us that the hex right next to the blue one we just marked has to be erased. Unfortunately, that only gives us a "?".

The Blue 2 overlay we just uncovered gives us our next clue. When we expand it, it covers both the -2- and the "2", as well as all of their surrounding hexes. Each of these still needs one blue hex apiece, and they will give the hub the two it needs. So the two orange hexes covered by this overlay that are not attached to either empty hex can now be erased:


Believe it or not, we can complete one of the diagonals now. From the upper-right corner, the diagonal headed by an "8" which begins with a "?" has been reduced to eight active hexes; we can mark all the others in the line. Doing so gives our -2- the second blue hex it needs, and we can erase its final orange hex.

We can't quite complete the second diagonal headed by an "8" extending from this section, but we can come close. The line contains nine active hexes, but two of them border the "2" we're trying to solve. Since only one of those can be marked, we can safely mark the rest, leaving this pair unsolved for right now.



We can't do much more at the top except erase the final orange hex from the "2" we uncovered on the top-left edge of the -2-.

We'll move to the bottom-left grid, then. On the upper-left edge of the "0" on its top-right corner is a "1" with only one possible blue hex to claim. The "1" below it shares the same blue hex, so we can erase its other two orange hexes. Doing that leaves the "1" below the "0" with only one choice for a blue hex, as well. Go ahead and erase the orange hexes from the empty "1" cells which share the blue hexes we just marked.

Now, expand the Blue 3 hub we just revealed. It covers five active hexes, but three of them surround the "2" to which the hub contributes its first blue hex. Since only one of those three can thus be marked, we can safely mark the other two that this overlay covers:




Over in the bottom-right grid, we just gave the "2" halfway down its first column a second blue hex. We can go ahead and erase the cell on its bottom-right edge, but we cannot immediately solve the "3" that comes up. Nor can we do much with the Blue 6 hub here; the overlay just covers too many active hexes.

We can, however, use the Blue 5 hub in the grid's second column for another clue. If we expand it, it covers three blue hexes, as well as the "3" we just opened and its remaining orange hexes, plus one additional orange hex. We already know that only one of the two hexes bordering the "3" can be marked, so we can safely mark that third orange hex the overlay covers.

The column in which this blue hex falls is headed by a "5", and it also became the column's fifth blue hex. We can now erase the rest of the column; as usual, erase the hexes from the "0" that appears, as well:





So now, we can work with the Blue 4 hub a little farther up. This overlay covers three blue hexes already; it also covers the "2" on its upper-right edge and all of its surrounding hexes, plus two more orange hexes. As we have seen, we'll get the hub's fourth blue hex from whichever one we mark to complete the "2"; erase the other two orange hexes it covers.

The "2" that we uncover from these eliminations is in a great position; it already owns two blue hexes along its left edge, so we get to erase three more. One of those erasures exposes a "0" that allows for two more eliminations yet. Now, the "3" we opened up on top of the "2" is surrounded by exactly three active hexes; we just need to mark the third, which also gives a second blue hex to the "2" positioned on the upper-right edge of the Blue 4 hub. We can finish most of the grid from this:

At the top-left corner of the puzzle is another diagonal headed by an "8"; this one contains an embedded line headed by a "4". We can solve the embedded line now; it's been reduced to only four active hexes. We just need to mark the last one at the end of the line.

The "2" on the upper-left edge of the hex we just marked gains its second blue hex from this. Erase the cell on the upper-right edge of the "2" now. The "2" that this uncovers has only one obvious second blue hex to claim, and that will give a shared blue hex to the "1" on its upper-left edge. We can then clear the cell on top of the "1", but we cannot solve the "3" that is revealed yet.


Returning to the bottom-center of this grid now, the "1" above the "0" has a blue hex. We can erase the orange hex to its left. This leaves its respective embedded line, headed by a "1", with only one remaining hex to mark to complete the whole column. When we mark it, the "3" on that hex's upper-left edge gains its third blue hex. Erase the cell below the "3". The "2" that appears shares its two blue hexes with the "3"; finish the rest of this section by following the relationships:

We should be able to finish the top-right grid now. Its first column, headed by a "7", now contains exactly seven active hexes; marking the last one gives that pesky empty "4" cell its fourth blue hex. Erase the final orange hex on top of it, then just complete the "5" that appears:

We're getting there! We're about 2/3 of the way through the puzzle now:



Let's go back to the Blue 6 hub in the first column of the bottom-right grid now; if we expand the overlay, it now covers six active hexes. Marking the last one also gives the embedded line headed by a "3" in the bottom-left grid its third hex. We can erase the last one from that line.

The "2" on the bottom-left edge of the Blue 3 in that line receives its second blue hex from this. Erase the remaining two orange hexes from the "2". We uncover another "2" below this one; it owns two blue hexes to its right, allowing us to clear two more. The "3" uncovered below this "2" has only three active hexes to work with; marking them completes several requirements, including the embedded line headed by a "1" down here:

We'll continue working from the "1" in the center of the embedded line we just completed, clearing the two hexes to its left. When we do so, the "1" at the top of the embedded line is left with just one possible blue hex to claim. In fact, this blue hex is shared by three empty hexes: a pair of empty "1" cells, and a "2". Go ahead and clear the orange hexes from the "1" on top of the blue hex we just marked.

We want to return now to the Blue 3 hub at the very bottom of the grid. When we expand it, the overlay now only covers three active hexes. Mark the last one. That blue hex is shared with both the "1" and the "2" along its top rim. This establishes a new sequence of relationships:




We can continue working towards the top of this grid now. The "2" on the bottom-right edge of the Blue 8 towards the left side has its two blue hexes, so we can erase the one on top of it. Now, towards the top-center of the grid is a diagonal line of three empty "2" cells; the center of these has only two obvious blue hexes that we can now mark. See if you can continue from here:


We've now reduced that big diagonal headed by a "10" from the top-right corner to 10 active hexes; we just need to mark the last one. Doing so gives the "3" at the end of that same line its third blue hex; erase the one on its top-left edge. That reduces the embedded line headed by a "6", which extends from the top-right grid, to six active hexes, letting us mark the last one in that line. However, we cannot completely solve the empty "4" we just revealed yet.

Aside from those stragglers in the bottom half of the puzzle, we're down to just the top-left grid now. We unfortunately have not been able to do much up here yet; now that we have solved almost everything in the bottom half, we want to re-examine the line headers to see if we can fill in any more gaps.

It turns out that the column headed by a "3" near the center of this grid has already been given its three blue hexes, letting us erase the top four hexes. The -2- we reveal at the bottom of that chain has two active hexes on the left and only one on the right. We need to mark the latter; that also gives a blue hex to the "1" on the bottom-right edge of the -2-. Just this one move establishes a lengthy series of relationships that fills in the final three columns of this grid:

We can now pretty much start at the top of the column governed by a "5" in the center of the grid, and the chain of empty hexes we uncovered at the start of the last sequence, and work our way down; after completing that column, work the next column to the left from bottom to top:




The two diagonals extending from the top-center of this grid can now be solved, as well. The line headed by the "5" contains five blue hexes; erase the rest of that line. The next line above, which is headed by a "7", now contains seven active hexes; mark the rest. This also, finally, solves those final three hexes in the bottom-right grid.

The "2" located three spaces down from the top in the grid's central column has only two possibilities for its blue hexes. Mark them both; one will be shared with the "2" below the first one, giving it a second blue hex. The other will be shared with the "3" above the "2", as well as the "3" on the top-left edge of this "3" (believe me, this is harder to write out than it will be to see in-game; just study the cells after marking those two, and it will become clear). This opens up a new series of relationships on the grid:

We can still carry out that series a little further. Start with the "1" at the very bottom of the grid's fourth column, and mark the only hex it can claim. That gives the "2" above it a second blue hex, allowing us to erase two more. On top of the "2", we reveal a second "2"; the "3" on the upper-right edge of this one is left with only an obvious third blue hex to claim. From that, see if you can get to here:

Fully six of the remaining nine orange hexes will be marked. We're going to finish the bottom-left grid now. Expand the Blue 6 overlay in its top-left corner; it now covers six blue hexes. When we erase the only orange hex it covers, we leave the embedded diagonal headed by a "2" with only one final hex to mark, also completing the grid.

We will get four of the five remaining blue hexes just from solving our final two diagonals at the upper-left. Each has been reduced to the same number of active hexes as blue hexes they require; simply fill in the rest within each line:





And we're left with only a small triangle of three orange hexes to work. In the third column of the grid, the "2" halfway down already has its two blue hexes; when we erase the cell on its top-left edge, we uncover a "3". The "2" directly below it is thus left with only one possible hex to claim for a second blue one. Mark it, then erase the final hex of the puzzle.



Honestly, there is nothing extraordinarily difficult about this puzzle; it's just extremely long and requires a lot of maneuvering back and forth between different sections. The hardest part is in determining when to use the overlays provided, and in keeping up with the relationships on the grid. The relationships themselves are a little more complicated in places, and the sequences are longer; thus, this puzzle is basically a marathon more than it is a difficult logic puzzle.

And with that, we have come to the hardest challenge Hexcells Plus has on offer. We'll continue next time as we prepare to finish the game.

Hexes Earned for Completing This Puzzle: 25

We are only two puzzles from the end of Hexcells Plus! As we reach the conclusion of the game, I want to thank you for joining me for this playthrough, especially if you also joined me for the original game. Puzzle 6-5 is the longest and meanest the game has to offer. Take a look!




This puzzle is so large that I have to use JPG image format instead of PNG like I have been to keep the file size small enough for Steam to let me upload! Also, due to the very large dimensional size of the puzzle board, I will have to break a convention I have used for annotations. Previously, I have tried not to overlap an annotation with the grid itself; this will be almost unavoidable for this puzzle. However, where this occurs, I will try to only cover up a section we are not ready to solve, or that has been completely solved and will not need to be revisited.

We're given a decent enough head start. Almost every conceivable line has been given a numbered header. In the center, we're given a Blue 5 hub and a pair of {3} cells. Two additional hubs are positioned in between clusters in the lower half of the board. And on the left, we're given a cluster of empty hexes and a Blue 4 hub.

Remember that no matter how large or complicated a puzzle looks, the fundamentals we have been using since the original Hexcells still apply. Nothing we do will deviate from the rules that have been established; only their application and the way we think through the puzzle may change. Here we go.

With so many line headers, it can be a good idea to at least run through them a couple of times to see if any action can be taken just from them. Note, however, that many of the lines are sub-divided; be sure to take the embedded lines into consideration, as well.

It isn't much, but over on the right side of the puzzle, there are two columns both headed by a {5}. The first one already contains a blue hex relatively high up; if we assume this is an endpoint, we can determine the maximum reach for the column's chain of five consecutive blue hexes. This allows the two bottom cells to be erased.

Now, for the second such column, we have a total of eight orange hexes. If we number them and run through the possible combinations for five consecutive blue hexes, we get: Cells 1-5; Cells 2-6; Cells 3-7; and Cells 4-8. The fourth and fifth hexes are included in every scenario; these are guaranteed to be marked.

One final step to start off with: In the center, below the Blue 5 hub, we have a pair of {3} cells in the central ring. They each have only four consecutive active hexes around them, so we can immediately mark the middle two in each ring. The only question is whether or not they will share their final blue hex at the bottom-center of the ring...



Okay, let's go ahead and start working on the cluster on the far left side. We can actually solve that -2- immediately; it may have four active hexes, but the blue one we're given is in the middle of that ring. So we need to simply erase the two orange hexes connected to the blue one, then mark the last one.

We uncover a pair of empty "2" cells from this; the one on the upper-right edge of the -2- actually shares two blue hexes with it. We can now erase the two orange hexes from this "2". That leaves the -3- with only three surrounding active hexes; now, just mark the other two.

The "2" under the -3- gets a second blue hex from this; erase the one on its bottom-left edge, leaving the "2" in the first column with only one left to claim for its second blue one.






We have two diagonals headed by a "6" up here; we can make one step within the lowest one. It is sub-divided, with a "5" for its Secondary Line Header. That means this segment can only contain one blue hex, which we have already marked. So erase the final hex at the end of this segment. The {2} that is uncovered is very helpful. It pops up with two consecutive blue hexes already bordering it, so just erase its two orange ones.

To work through the next sequence, we need both the remaining Blue 4 overlay and the upper of our two diagonals headed by a "6". This particular line is sub-divided such that it needs two blue hexes at this end; we don't have any yet. Three of the four orange hexes at this end are also covered by the overlay.

The overlay also covers two blue hexes. We can see relatively easily that its third will come from the "2" on the upper-left edge of the hub; after all, the hub gives it one blue hex, and its two orange hexes are both covered. The problem, though, is that the "2" on the hub's upper-right edge also shares one of these orange hexes with the first "2". Are we marking the shared hex and then marking the hub's fourth blue hex from somewhere else?

Let's re-examine our diagonal. With three of this segment's four orange hexes covered by the overlay, and a requirement that two be marked, it's impossible to satisfy the line's requirements without marking one inside the overlay. Additionally, the restrictions imposed by the overlay ensure that only one of them can be marked. This makes the line segment's fourth orange hex a guaranteed blue hex; our next step, then, is to mark this one:



We've now given the "1" on the edge here a blue hex; erasing the cell above it reveals a -3- with only four continuous active hexes around it. So when we mark the second endpoint, we can complete this line segment by simply erasing the final hex.

This now gives the "2" on top of the {2} its second blue hex. When we erase its last one, we leave the "2" on the upper-left edge of our hub with only one choice for its second blue hex:






Let's go ahead and use that Blue 6 overlay now; expand it, and it will cover exactly six active hexes. So just mark the other three it needs. Notice now that the puzzle's second column is headed by a "2", and that we just gave it a second blue hex. So to finish this sequence, erase the rest of the column:



The dynamic of a -2-, a "3", and a "1" that we just revealed is particularly interesting. We actually can solve the -2-. The trick is where the "3" is sandwiched between the -2- and the "1"; we clearly can't give the "3" its complete set of blue hexes without sharing one of them with the "1". That means only one of the hexes along the bottom rim of the "3" can be marked, guaranteeing that both hexes along its top rim will be. And guess which empty hex those two blue hexes will be shared with? So now, clear the cell on the upper-right edge of the -2-. Then, clear the hexes from the "1" that this reveals.

We've made a decent dent in this puzzle, but we still have a long way to go!



The -2- that we just uncovered is actually in a great position. We have two active hexes along its bottom rim, one of which is already marked. So we just need to erase the second hex in that pair and mark the only other active hex on its top-right edge. The empty "1" cells now positioned above and below the -2- get their blue hexes from this; erase their remaining orange hexes. The last hex in this small grid will then be marked to give the "3" its third blue hex.

Solving that last cluster also lets us solve the column headed by the "5" here. We've just reduced it to five total active hexes, so just mark the rest.

Back up at the top now is a diagonal headed by a "9", which is sub-divided such that this end of the line can only contribute three blue hexes. We already have two; two of the three remaining orange hexes at this end border that -3- we haven't been able to solve yet. Since the -3- has no other orange hexes, we know one of these two has to be marked. That means the final orange hex in this line segment can be erased.

Well, that gives us a second -3-; we can do little more right now than mark the second endpoint in its ring.







So let's take the diagonal right above this one, which is headed by a "5". It, too, is sub-divided; the Secondary Line Header is "3", meaning that only two blue hexes can come from this section. The hex we just marked gave it the second; we can now erase those in between the blue ones.

Those erasures leave the -3- we uncovered a few moments ago with only three active hexes; when we mark the last one, it is shared with the first -3-, and we can erase its final orange hex.

We also get two more blue hexes from this setup. First, we can give the "3" we just revealed the only other obvious blue hex it can claim. Secondly, we have now left the column headed by a {2} with only three consecutive active hexes; so now, we can mark the middle one in that chain.




Now, near the bottom-left corner is a Blue 2 hub just kind of floating there. If we expand its overlay now, it covers the required two blue hexes. That lets us erase a couple more from the narrow strip at the bottom. This leaves that pair of empty "1" cells in the corner with just one obvious blue hex to claim.

We'll need that Blue 3 overlay just about halfway up this side now. Expand it; it covers one blue hex and three orange hexes. Naturally, we need to mark two of these. Notice that two of them fall into a column headed by a "4", which already contains three blue hexes. Since only one of those can be marked, the overlay's third orange hex is guaranteed to be. Marking it does a couple of things. First, it provides a second blue hex for the column headed by the {2}, letting us erase its final hex. Secondly, we can now erase any hex in the column headed by the "4" that is not covered by this overlay. We even get two zeroes to clear out, leading to a few new empty "1" cells claiming obvious blue hexes:



One of the hexes we just marked for those empty "1" cells gave a second blue one to the "2" at the bottom of the column headed by the "4" that we have been working on. We need to clear the remaining orange hex from this "2". This reveals another -3-, but it has the same pattern of four continuous active hexes. So mark the second endpoint as usual. Guess what? This column, which is headed by a "2", now has its second blue hex! Erase the rest of the line:

It may not seem like it, but the Blue 5 we uncovered a few moments ago is actually very useful here. Let's expand it next. It covers three blue hexes already. While it covers a lot of orange hexes, we only need to worry about four of them: The two beside the -3- we still need to solve, and the two on the left side of the "2" on top of the hub. These are the ones also covered by that Blue 3 overlay, and from which one more blue hex is already guaranteed to be marked. So all of the other orange hexes inside this overlay can be erased; one of them opens up a "0" that gives us one additional erasure, too:

The "2" under the "0" has only two obvious hexes to claim, and we'll also complete that -3- when we mark them. But we still need something else to complete these two overlays...

We are working now in a column headed by a "5". It has three blue hexes already; we can use our modified counting system again to identify two pairs of hexes from which the remaining two will be derived, and that will let us clear a vital hex from this column. First, at the top of the line is an adjacent "2" whose only choices for a second blue hex fall into this column. Secondly, right above where we're working is a "1" whose only two choices for a blue hex fall into this column. The hex right below this pair has to be erased:

The "2" that we just revealed is extremely important. Both empty "1" cells along its bottom rim are now left with only obvious blue hexes to claim. When we mark both, we also give these two overlays the last blue hex they need and can erase the last orange hex they still cover.




While we're here, let's examine the Blue 3 overlay at the bottom of this section. This one now covers three blue hexes, allowing us to erase three more. One of them opens up yet another "0" to follow, but we can't complete the empty "1" cells revealed yet.

An important clue we do receive, though, is that we reveal a "1" in a little triangle of hexes above this grid. Notice that its only two orange hexes fall into an embedded diagonal headed by a "1". Since we have to mark one of these two, the rest of the line can now be erased:




So at the end of the line, we have the following empty hexes: "2", "2", "3", "2", and "1". The first "2", the third "2", and the "1" all have obvious blue hexes to mark. Mark them now.

We need another clue to solve the "2" and "3" in the center of that chain. That clue comes from the next line down, which is governed by a "7". We've reduced it to seven active hexes; when we mark the rest of the line, we'll be able to clear the final hex from this cluster:

Let's take two final steps here, then. First: Remember the pair of empty "1" cells we revealed from the zeroes a little bit ago? The top one now has a blue hex. When we clear its remaining orange hex, the lower "1" is left with only an obvious hex to claim. Marking it gives a third blue hex to the "3" we just revealed, letting us clear one more hex. The "3" that it opens pops up with three blue hexes; erase the last orange hex of this section.

The final step for this sequence involves that Blue 3 overlay. It expands to cover three blue hexes; when we erase the sole orange hex it covers, we'll be able to solve that "1" in our other little triangle of hexes:

We're about a third or so of the way finished now; here's the setup for the next sequence:



Two more lines can be solved now. First, we have an embedded line headed by a "5" right above the one headed by the "1". It now contains five active hexes, so mark the rest. Secondly, we can solve one of the columns headed by a {5}. When we solved that little triangle of hexes a moment ago, we gave this one its second endpoint in the chain (the first was given to us). So now, just fill in the rest of the chain and erase the hexes at the top of the column:

We haven't done anything with the top-left section in quite awhile, so let's see if we can expand towards the center. We still have two overlays up here; start with the Blue 6. It now expands to cover six active hexes. Mark the last one, then expand the Blue 4 overlay above. It will now cover four blue hexes, allowing us to erase the orange hex it covers. That {2} we uncover from this has only two active hexes to worry about, so just mark the second.

That Blue 3 overlay is a dead end; yes, it covers two blue hexes, but there is simply not enough information to determine which of its many orange hexes it will claim. So we will need to come back.

There are a couple of places we can go now; one was actually opened to us in the beginning. The other, at the top, we opened up upon completion of the column headed by the {5}.

Let's examine the route we opened up at the start. Remember when we marked the center hexes of the other column headed by a {5}? One of them falls into a diagonal governed by a "6", which is sub-divided such that only one blue hex can come from this section. So we can actually erase the rest of this line segment.

This didn't do nearly as much as we might have hoped; let's try the other path. We uncovered a "1" and a -3- at the top of the column headed by the {5} that we completed earlier. The trick to this pair is that we cannot complete the -3- without giving a shared blue hex to the "1"; we just don't know which one yet. So the hexes on the bottom rim of the -3- are guaranteed to be marked; the two along the top rim of the "1" are guaranteed to be eliminated. One of them gives us a "0" to follow. When we complete the sequence that it begins, we'll be able to complete the -3-:

Let's go ahead and use the Blue 3 overlay at the top-right of this section. It expands to cover exactly three active hexes; mark them all. In doing so, we can work a little bit on the diagonal headed by an "11" at the bottom edge of the overlay. This one is also sub-divided; the Secondary Line Header is an "8", meaning that only three hexes can come from this end. We just completed that requirement, so erase the final hex within this segment of the line.

We uncover a -2- that we can solve like so many others, so go ahead and do that next. From here, though, move to the other Blue 3 overlay. This one already covers two blue hexes; the third will come from the one we give to the "2" right above the hub. Erase from the overlay the hexes not connected to the "2":



The "2" we just revealed on the bottom-left edge of the Blue 3 has its two blue hexes. When we erase the one above it, we leave the "2" on top of the hub with only one left to claim. When we mark it, we feed a blue hex to the {2} at an endpoint, telling us exactly how to solve it.

Now, we have a Blue 2 hub; if we expand its overlay, it covers one blue and two orange hexes. The trick here is that the two orange hexes fall into a column headed by a "5". The line is sub-divided so that only one blue hex can come from this end; since it has to be one of the top two in order to complete the overlay, we can safely erase the third hex down this column.

And now, we get another -2- with three continuous active hexes. Solve it as usual; it will tell us how to complete this end of the column:







We just uncovered a "1" at the top of the column, with a Blue 4 under it. Erase the hex on the bottom-left edge of the "1". Now, expand the Blue 4; the overlay now covers exactly four active hexes, so mark the other two.

I'll go ahead and end this sequence now; we'll need two overlays in tandem to complete the next step...







We now have a Blue 3 and a Blue 4 in close proximity to each other, with a decent number of orange hexes between them. If we expand both individually, we find that they each need exactly one blue hex apiece; and if we expand them simultaneously, we find that they share four orange hexes between them.

Since the Blue 4 overlay covers fewer orange hexes, it gets priority. One of the four orange hexes it covers will also be used to complete the Blue 3 overlay. As such, the hexes covered by the Blue 3 overlay that are not shared with the Blue 4 have to be erased:




The "1" that opened on the bottom-right edge of the Blue 3 lets us clear two more cells. Likewise, the "1" at the top of the column headed by the "6" lets us clear another, leaving only one for both overlays to claim.

We'll shift back to the top-right side of the puzzle now. We need to work on the other Blue 4 overlay up here. Expand this one, and it covers three blue hexes and five orange hexes. Three of the orange hexes fall into a diagonal governed by a "4", which is split up so that only two blue ones can come from this end. We have one already; these three orange hexes are the only ones left to pick its second. As it will make the overlay's fourth blue hex, the two orange hexes it covers in the next line down have to be erased next:

This also helps us somewhat with our other column headed by a {5}; we've effectively cut off its chain of five blue hexes at the top of the column, so we can go ahead and erase the very top hex. This also leaves the column with only six active hexes; the only thing we don't know is where the endpoints are. The four center active hexes need to all be marked now.

The "2" we just revealed at the top of the column has only two active hexes to claim. Marking both also gives the second blue hex to the top segment of our diagonal headed by the "4". We can erase the last hex from this segment, leaving the "3" on that hex's bottom-right edge with only three active hexes. Mark the other, also giving the "3" on the upper-right edge of this one a complete set of blue hexes. Go ahead and erase the final orange hex from this "3".

So it turns out that when we filled in the hexes for all of those empty "3" cells, we marked the endpoint for our chain of hexes in the column governed by the {5}; so now, erase the hex at the bottom of the column, then follow the "0" that appears. We'll be able to solve all but four hexes in the bottom grid from this:

It's a slow process, but we are gradually wearing it down:



We have a Blue 3 overlay down here now; we might as well solve it next since it only covers three total active hexes.

We'll return now to the upper-right section. We have two diagonals each governed by a "9" here; the upper one is divided to require only two hexes at this end, which it now has. That lets us erase two more. We get a "1" at the end of this segment, with a blue hex on its upper-right edge; clear its two orange hexes.

We now need to look at the lower of these lines. This one needs three blue hexes at this end; the Secondary Line Header is a "6". There are four total active hexes, one of which is blue. Two of the three orange hexes are sandwiched in between a pair of empty "2" cells. For the "2" at the beginning of the line above, they are its only choices for a second blue hex, guaranteeing that only one of them will be marked. The third, then, is now a guaranteed blue hex.

This gives the "3" above the hex we just marked its third blue one; when we erase its final orange hex, we'll have enough information to complete this line segment, as well:






Look now at the Blue 7 hub we revealed a moment ago. If we expand it, it covers six blue hexes already and needs just one more. We know that it will come from the final blue hex we give to the "4" under the hub; the only other orange hex it covers can now be erased. We'll be able to solve that "4" by marking the endpoints around the -3- that appears from this erasure.

The "2" we just revealed under the "4" will share two blue hexes with it. Erase the hex on its bottom-right edge. That reveals another "2"; we'll look at that other Blue 7 overlay next.






Okay, expand that other Blue 7; this overlay only covers four blue hexes. It also covers four orange hexes.

We can't use the same logic to narrow down the hexes as we have used before. We can see that the "2" and the -3- here share an orange hex between them. The question is whether or not we can feasibly mark that one. The restrictions imposed by the overlay say that we cannot, simply because it would erase too many hexes. So we have to erase that one and mark the other three:


Let's use the Blue 6 that we just uncovered; this one expands to cover six active hexes, so again, we just need to mark the rest.

We actually uncover another Blue 6 in another triangle of three hexes in between grids. This one is in the exact same position, covering only six active hexes. This time, though, we'll also feed the fourth blue hex to the "4" at the top-right corner of the lower grid that we had to abandon earlier. We'll then be able to erase the final hex in that grid:


This leaves only the center to finish up. We'll be able to solve another line now. We still have another diagonal headed by a "4" from near the top-right; we've now given it the four blue hexes it needs and can erase the rest of the line.

We're almost at the end now, which means it's going to start getting tougher. At this point, it seems like almost everything left is one or two away from what we need to open up the solution. Where do we go next?

Let's expand the Blue 3 overlay in our triangle of hexes between the center and bottom grids, towards the right side. It already covers two blue hexes, which is a big help. And of the orange hexes it covers, two border the empty "3" cell we uncovered just moments ago when we solved that last line. Since the "3" has only four total active hexes around it, we clearly cannot complete it without marking one of these two. So any orange hexes the overlay covers which are not tied to the "3" can be cleared next:

By themselves, the "?" cells do nothing for us. However, our previous analysis of the Blue 3 overlay also tells us that the two orange hexes bordering the "3" which are not covered by this overlay are guaranteed blue hexes. They will be shared with the "2" on the upper-right edge of the "3", as well. When we mark them and then clear the other two hexes from the "2", we leave the "2" on the outer edge with only two obvious blue hexes to claim.

That sequence lets us solve the diagonal governed by a "6" from the right-center of the puzzle. We now have six blue hexes for the line; when we erase the final one, we can also solve that empty "3" cell.





We do get one more column to complete from this. It's the one headed by a "3" not far right of the center. It now contains three blue hexes, allowing us to erase three more.

If we now expand the Blue 5 cell in the middle of that column, we find that its overlay covers five blue hexes; we can clear the two orange ones it covers. And doing this, in turn, allows us to solve the embedded line headed by a "6" that runs through this section; it is now left with six active hexes, so we just need to mark what's left:



So this solves enough that the Blue 7 hub here expands to cover exactly seven active hexes. When we mark the last one, it gives a second blue hex to the "2" at the top of the section, just beyond the maximum reach of the overlay. Clearing the hex on its top-right edge leaves another "2" with only two active hexes to worry about marking.

We can now solve both columns governed by a "4" running through this cluster. The one on the left contains exactly four actve hexes and needs its last one to be marked. The one on the right has four blue hexes and needs two to be erased.





That gave us more to work with at the bottom of the section. The "3" we just revealed has only three active hexes; marking the other two will give the "2" right above it a second blue hex. We can't immediately solve the new "3" that appears after erasing the final hex from the "2", however.

For that, we'll need that Blue 5 hub along the right edge that we have done nothing with so far. We now have some information to at least get an elimination from it. The overlay already covers four blue hexes; it also covers the only two remaining orange hexes this "3" can use for its third blue hex, plus one more. That additional orange hex is our next erasure.


The other Blue 6 overlay in this section will let us solve those last few cells. That last erasure reduced its overlay to cover six active hexes; marking the last one gives that "3" its third blue hex. When we erase the final hex from the "3", we reduce the column in which that hex was positioned to five active hexes. As the column is headed by a "5", we just need to mark the final hex in this section:

We are down to the final section in the game's ultimate puzzle:



We're left with no obvious direction on how to proceed. If we examine the remaining lines, they seem to all still have too many active hexes to narrow down which ones to mark; the same goes for the remaining overlays.

It may take a lot of scanning, but there are a few moves we can make. The first begins with our embedded line governed by the "7", which passes through one of the {3} cells in the center. The segment contains eight active hexes, including both orange hexes from which the {3} will claim its third blue hex. We know only one of these can be marked now; this means the other orange hex is guaranteed to be marked, and this will solve our final triangle of hexes between the center and bottom grids.

The Blue 3 hub we reveal here gives us another elimination. Expand it, and the overlay covers three blue hexes; erase the only orange hex it covers:






We can also complete the embedded line headed by a "3" from the top-left of the central cluster, near the Blue 2 hub sandwiched in between grids. This one contains three blue hexes now; erase the rest of the line.

Sadly, we get nothing but "?" cells from that, so we can't build off of it. What now?

Let's look at the Blue 2 overlay several columns to the left of the {3} cells. It expands to cover one blue hex and a slew of orange hexes. Three of those fall into an embedded line headed by a "5", which contains three blue hexes. There are only four total active hexes remaining within the line; clearly, one, but only one, of these three has to be marked. As a result, the first hex of the embedded line is now a guaranteed blue hex; we can also erase from the overlay its orange hexes from the next line above:

We can complete two more columns on the left side of the puzzle from this. First, the column headed by a "5" furthest left now has five blue hexes; erase its final hex. Secondly, the one headed by a "4" three columns to the right has been reduced to four active hexes; mark the last two.

After marking those last two hexes, we also give the embedded line headed by the "5" that we're trying to solve its final blue hex. This erases two more cells. It also reduces the column headed by the "6" to six active hexes; mark the rest to also solve all of the nearby empty hexes. Be sure to erase the final orange hex in this cluster; the "2" below it will get its second blue hex from this sequence.

Now, we can solve that other Blue 2 hub. It expands to cover its two blue hexes, so erase the only orange hex it covers.

This ends up being a very important erasure; the cell is the beginning of an embedded line headed by a "7", and we've just reduced it to seven active hexes. This gives us two more to mark. Afterward, the full-length column headed by a "4" just right of center will gain its fourth blue hex, letting us erase the rest of the line.

Now, the embedded diagonal headed by the "8" here can be solved as we have reduced it to eight active hexes. This gives us two more blue hexes to mark:






We're down to the final four blue hexes! We can actually solve our pair of {3} cells now. A little bit ago, we erased a hex from the one on the right, leaving it with only three active hexes. This means we'll mark the shared hex between them and erase the last hex from the {3} on the left. And when we do this, we can go ahead and solve the Blue 5 in the center; the hex we just marked was the fifth one it needed, so we'll be able to erase the final orange hex the overlay covers.

We end up being able to solve most of the remaining lines now. I'm going to let the next screenshot explain this as it will be much easier to see graphically; in essence, though, we'll end up erasing almost everything that's left, and marking only one additional hex, that one being within the embedded diagonal headed by a "2" at the top-right of the grid.



We're done!!!!!! That last sequence left the puzzle with only two active hexes, and we have only two blue hexes left to mark. Mark both to defeat the giant!!!



And that's it!! The longest and most difficult puzzle in Hexcells Plus has gone down in flames! We're only one puzzle away from completing the game and unlocking the "Perfectionist" achievement for a second time.

The difficulty of this puzzle lies primarily in its size and the fact that there is a lot of jumping around. There are also sections where it might seem like we should be able to solve them but in fact need more information, forcing us to abandon them until later. Also, with so many line headers and overlays which eventually open, it can lead to information overload.

While writing this solution, I realized there may be some places during the second half where a few steps could have been worked in a different order for a "cleaner" solution. I'm electing not to change it, though. While this approach may not be perfect, it works, and that's one of the things about Hexcells. It may take a long time to find the way forward, but it's there, and you don't have to play flawlessly to get to the solution. As always, I encourage you to work through it at your pace and your skill level; this is just my approach for this particular guide, and you may do things differently. The only thing that matters is that the approach you take works for you!

All of that said, join me in the next puzzle as we wrap up the game!

Hexes Earned for Completing This Puzzle: 25

Whew!!! After the mammoth that was Puzzle 6-5, we have now reached the final stage of Hexcells Plus. If you've played along with me using this walkthrough, thank you!


Well, it looks like we're done! So now, we just mark all of the hexes and...



Wait, what's this? You mean we actually have to solve it this time??

For its size, this puzzle contains a lot of line headers. Also heed the hints at the bottom of the screen as you work through it. To start with, an English lesson:

"I'll take English and Grammar for $200, Alex."

"Answer: 'A,' 'E,' 'I,' 'O,' 'U.'"

"What are 'vowels,' Alex?"

"Correct!"

Humor aside, the word "FINISH" obviously has two vowels in it, and the game is telling us that between them, only five of their total 10 hexes will be marked. Additionally, one of the letters will be nothing but empty hexes. Finally, between the "F" and the "H," we will mark a grand total of 16 blue hexes.

With a small puzzle like this, it's a good idea to look at the line headers to see if any lines can be completed outright. In this case, five can be completed. Two are empty columns; two are headed by a "5" and contain only five hexes apiece; and one is headed by a "2" and contains only two hexes. What we have after working them is this:


In doing this, we actually mark 12 of the 16 total blue hexes required for the "F" and the "H." The first column of the "F" is headed by a -4-, and we haven't done anything with it yet; this means that the last four hexes to complete these letters will come from this column. As a result, we can erase the last two hexes in the "H" and the two center hexes of the "F."

Now, notice that each hex in the first column of the "F" marks the beginning of a horizontal line extending across the puzzle. The lines headed by a "6" and a "7" near the middle are sub-divided at the second "I." Each of these segments is headed by a "3," so each line will need three blue hexes to come from the "I," "S," or "H."

In the center of the "S" is an empty "1" cell, which falls into one of the embedded lines. This segment also happens to contain two blue hexes already. The "1" has only two active hexes to work with, to its left and right. The embedded line's third hex will naturally also come from one of them, meaning that we can erase from the second "I" the hex falling into this line segment:


Erasing the middle hex from that "I" gives us a very important clue. We're told that one letter will not have any blue hexes whatsoever. If we look at each one, we can see that all of the consonants will have blue hexes; they each have a column header which mandates this. Only the two vowels have no column headers (as the horizontal lines cross multiple letters, they cannot be used as a meaningful indicator of which letters will or will not contain blue hexes). And we've just erased a hex from the second "I."

This is what helps blow the puzzle wide open. For the sake of argument, let's say we could mark the other four hexes within the second "I" and mark just one in the first. We break our rule that one of the letters has no blue hexes, since we would now specifically be giving each letter at least one blue hex. As a result, this one elimination tells us that the second "I" is empty, and the first "I" will contribute all five of the blue hexes required by the vowels:

Remember that we have two embedded lines, each headed by a "3", that we need to resolve. With the second "I" fully erased, we've reduced the top such line to exactly three active hexes; marking the last one gives us a blue hex within the first column of the "S," which itself is headed by a {2}. Only two hexes are linked to that blue hex as possible candidates for a second capture for the column; as a result, we can erase the bottom hex of the column. Once we do this, we leave the bottom horizontal line, headed by a "7", with seven active hexes. Mark the rest:

Now, in the final column of the "N," we have marked a blue hex at the endpoint. This column is also headed by a {2}; mark the next hex up, then erase the rest.

Our horizontal line governed by a "6" from near the top of the "F" has now been reduced to six total active hexes, including the ones we solved within our embedded line. When we mark the other two, we give a second blue hex to the first column of the "N," which is headed by a regular "2". We can now erase the rest of that column, as well:


There's a minor trick to this next sequence. For our horizontal line headed by the "7", we still can't solve the embedded line; however, we do know that the first segment of the line has to contain four blue hexes. The segment between the Primary and Secondary Line Headers has now itself been reduced to just four active hexes, giving us two more to mark.

Marking these two hexes gives a blue hex to the column headed by a "1" at the center of the "N," so erase the second hex here. Afterward, return to the first column of the "F," which now contains three blue hexes. Notice that if we mark the second hex up from the bottom, we'll actually link all of the blue hexes together, which we do not want in a column headed by a -4-. So to complete the column, erase that hex, then mark the one at the top:

We can now solve the two horizontal lines which remain. Each is headed by a "5". The top such line now contains five blue hexes, so erase its remaining orange hexes. By contrast, the lower line contains only five active hexes, so mark the last one.

When we do this, we can solve those last two columns. The column headed by a {2} is left with just two active hexes. Mark its last one, also allocating the puzzle's final blue hex. Erase the final two cells to complete the puzzle, Chapter 6, and all of Hexcells Plus!!!!!!!!!!!



Hexes Earned for Completing This Puzzle: 25

Wow!! What a ride this game has been!! Hexcells Plus is certainly a more intense game than its predecessor. As early as Chapter 2, the difficulty begins to ratchet up. The difficulty curve is noticeably steeper than in the original Hexcells. What keeps it interesting is the addition of what will become the series' final puzzle-solving mechanic in the form of Grid Overlays. It becomes critical to understand what they will and will not do for you in order to successfully utilize them.

When I first played through this game, I remember making mistakes as early as the later puzzles in Chapter 1! The logic involved in a lot of the puzzle solutions is more intricate and requires you to really pace yourself to make sure you're taking in everything before making a move. Some of the final puzzles gave me fits trying to perfect them, in part because I had so much trouble understanding just how to use overlays. My hope for this guide, just as it was for the original, is that it helps you to become a better player so that you can play through the puzzles yourself and use it as a resource if you need a hint. Of course, I've tried to design it with that in mind, as well as for anyone who wants a more casual experience and wants to step through most of the game just reading and absorbing the solutions.

It's also interesting that, instead of just being given the last puzzle, we actually have to solve a reasonably challenging puzzle to wind down the game. It's certainly not as hard as much of Chapter 6, but it gives us a couple of brain teasers to puzzle through before we can declare final victory.

And with that, we bring Hexcells Plus to a close. Whether you're just joining me or have followed me since the original Hexcells guide, I thank you for letting me be your guide through now two games' worth of puzzles. Be sure to also join me for Hexcells Infinite as we conclude the trilogy. Thank you for reading and for all of your support through the production of these guides. Take care, everyone.

Additional items not included as part of the main guide may be added here in the future. For now, consider this section to be a change log for any significant revisions which may be made later.

Saturday, May 9th, 2015: Version 1.0 published (original release)

Sunday, May 31st, 2015: Version 1.5 revisions completed:

Puzzle 4-1: Rewrote part of the solution by slightly expanding the discussion of Grid Overlays.

Screenshot edits/redos in the following puzzles:

• Chapter 1: Puzzles 1-1, 1-5, and 1-6
  
• Chapter 2: Puzzle 2-3
  
• Chapter 3: Puzzles 3-3 and 3-5
  
• Chapter 4: Puzzles 4-1, 4-2, and 4-4
  
• Chapter 5: Puzzles 5-2, 5-3, and 5-5
  
• Chapter 6: Puzzles 6-1, 6-2, 6-3, 6-5, and 6-6

Most of these just required minor rewriting or typo correction of annotations; a few required additional highlights to be added.

I also corrected grammatical errors and typos, and rewrote small sections for clarity, throughout the guide. However, none of the puzzle solutions required any major rewrites; as such, the vast majority of the guide remains unchanged.

Unless anything major is found later on, this should be the absolute final revision for this guide. On the whole, as I was editing it, I found very few major issues compared to the original Hexcells guide, and I think it is now just about as close to being perfect as I can get it. Please let me know in the comments if you find anything that needs to be fixed.

Thursday, December 10th, 2015: Version 2.0 revisions completed:

This update revolves primarily around editing the image code as I did for the original Hexcells guide to try and improve loading times. However, this was not as successful as it was for the previous guide. Your mileage may vary, but at least on my computer, the Steam Web renderer still locks up when I attempt to scroll through the guide, forcing me to restart the client altogether. My suspicion is that the number of images is having just as much an impact as the sizes of the images themselves. Until Valve updates the Steam Guides feature to handle more detailed strategy guides, there's very little I can do about this. I will update the Hexcells Infinite guide as planned, but I strongly urge you to view these longer guides in an external Web browser instead of within Steam itself. I apologize for this inconvenience.

Aside from editing the image code, I made a few very minor edits to the text, but these were so small as to not warrant any further detail. I did reupload one image for Puzzle 4-6, adding a couple of highlighted regions I had originally neglected to include.

As always, please comment if there are still significant issues with the guide itself that I have missed to this point. I would also appreciate it if you would let me know if the changes have helped in any way. It's possible it's just my computer continuing to experience loading issues.