I see that a few of the people who have written solvers have posted in their repositories counts of the number of solutions for some of the puzzles, such as the infamous 2025-09-15 hard, which had 2 764 800 solutions.
That's how many my dumb brute force solver counted for that one too, so it looks like we are all counting solutions the same way.
This raises a question.
Here's one solution to that particular puzzle
6/5(15,16) 1/2(23,22) 4/5(0,4) 1/5(2,1) 5/3(5,6) 2/5(7,8) 4/4(9,3) 0/0(17,10) 4/2(19,18) 3/3(12,11) 3/4(21,20) 5/5(14,13)
where the notation P/Q(A,B) means the tile that has P pips on one half and Q pips on the other have is placed so the P half is on square A (counting the leftmost square on the first row as 0, and then going left to right, top to bottom) and the Q half is on square B. The order the halves of a tile are given is the order they are in the puzzle specification, and the order the tiles are listed is the order from the puzzle specification.
My solver considers two solutions different if they do not produce identical strings when the solution is written in the aforementioned format.
I'm reasonably sure that this counts some solutions as different that most humans would count as the same.
For example supposed there is a 2/3 tile this is entirely inside a region that has to sum to 10. Another solution that is identical except that tile is rotated 180 degrees would probably be counted as the same solution by a human but as different by my solver.
Similarly, if there is also a 1/1 tile entirely inside that region, the 1/1 tile and the 2/3 tile could be swapped and my solver would say that is a different solution, but I think most humans would not.
How far does this go? Would a human tend to think of all permutations and orientations of a set of tiles that are all contained in the same constraint region as identical solutions?
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