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Also: did you notice that "perfect" potions don't actually have to be perfect? In week 4 you can throw togerther 40A, 3x64/48/32ABE, 2x16/48BE, 6A, 20B, 4AB, 24/8AE for a combined 200/266/200 perfect sight enhancer that sells for 5x1113 before stars.
> but like you said 3:4:3 recipes are still hard to find
...which is probably why enhancers and cures on average sell for a lot more than the basic potions.
If developers were evil, they would make those single magimin ingredients rare and expensive, and the most basic potion to have 3 magimin ingredients. Instead, they may have seen the difficulty of those and made rarest to have 4 or 5 magimin to become slime food.
I haven't finished proof yet, but I think 2 magimin recipe solutions are not that hard problem. Because that awfully sounds like the special case (2 dimension) of lattice problem, which is a P-problem. Gauss did proof and made algo... before that was the concept... ... free ref [sites.math.rutgers.edu]
In that original algo by Gauss does not have more then 2 'ingredients' so to speak, but seems easily modifiable.
My approach does not depend on finding the complete solution. I'm using a DFS search tree but with a couple of heuristics:
1) Consider recipes with the highest magi ingredients first. This makes high-magi recipes much more efficient to find.
2) Ignore all ingredients that don't contribute to the magi ratio, aim only for perfect ratio potions.
3) Ignore recipes with ingredient counts less than cauldron limit - 2
4) If at least 1 recipe is found after x nodes explored, end the search
5) Overflow is allowed, all solutions within a range of magi counts are considered (you may notice some of my recipes have slightly different magi counts).
With these heuristics I can generate near-optimal week 4 recipes in a few minutes (but like you said 3:4:3 recipes are still hard to find). The reason I haven't published week 4 is because I havent discovered all of its ingredients yet and i'm a bit lazy to play :P
I do not think you can get week4 recipes in time. Not because you are incapable, but because the problem you are solving is NP-complete. In plain english, compared to week 3 solution, week 4 solution will take 20,000 to 200,000 times more to compute.
What I suggest instead is get 8 ingredients solution/near-miss for Masterwork quality, and then add 2 or 3 ingredients that do not hurt balance of magimin. It's not the perfect answer, but it is one of the closest. It might be tricky to get 3:4:3 ratio, but its something much much more simple than that of the original problem.
Also, the amount of magimin does not correlate with money. Instead of finding exact magimin count answers, try to find solutions with the overflow of magimin count. This also ease the problem and you may find better answer in terms of profit faster.