Every single puzzle done in "INFINITE" mode counts towards the "60 down 999,999,940 to go" achievement, regardless of whether it has already been solved before. So the fastest way to get the achievement is to do random puzzles until you come across a super easy one, and then just solve it over and over. I came across such a puzzle, which I was able to eventually solve in just 20 seconds with a few rounds of practice (I went for a no-mistakes approach to make the monotony more fun):


The seed for this puzzle is 52530353 (< copy the text and use the "PASTE FROM CLIP BOARD" menu option for ease):


The puzzle requires you to place 36 blue hexes and the hints that get uncovered as the puzzle progresses are very obvious. For the lazy, here's the puzzle and its solution:



Once the puzzle has been solved, just click "RETRY" and do it again and again. There's no need to go back to the menu and re-enter the seed:


After a few rounds, you should have the puzzle almost memorized and as you can see, I was able to get the solve time as low as 20 seconds even, but more consistently in the mid-20s. And that's it! If the main game was enough for you and this achievement seems more like a chore but you really want that 100%, this is the fastest, easiest way without doing some cheesy auto-clicker tactic. Enjoy!