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The Mensa cube is a puzzle in which a solid cube has been partitioned into $N=11$ rigid parts. The goal of the puzzle is to re-assemble the cube from its parts and place it back in its rigid box. See pictures below.

How many non-rigid-motion equivalent ways are there to re-assemble the cube from the given parts?

I've found some formulae like Kasteleyn's 1961 count of domino tilings of checker boards, but nothing that would address the more general shapes of the Mensa puzzle. My personal suspicion is that there is only one way to solve it because of the lack of symmetries, but I have no idea how I would prove something like that.

I originally asked elsewhere and commenters recommended posting here (https://math.stackexchange.com/questions/4725630/how-many-ways-are-there-to-solve-the-mensa-cube-puzzle).

An example of the cube is https://etail247.com/products/mensa-original-and-best-box-cube

Here are pictures of the pieces:

mensa puzzle pieces 1/4

mensa puzzle pieces 2/4

mensa puzzle pieces 3/4

mensa puzzle pieces 4/4

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I used a computer to search for all solutions, and the number of solutions is

22

Here is a picture of the solutions, with the top layer on the left, bottom layer on the right.

Mensacube solutions

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  • $\begingroup$ It seems unlikely you wrote code from scratch to execute abstract algebra to find the solutions, so I'm curious what application did you use? Is this something Mathematica handles? $\endgroup$
    – J D
    Jun 28, 2023 at 13:10
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    $\begingroup$ @JD My Polyform Puzzle Solver can solve many packing puzzles. Unfortunately it has limited 3-dimensional capability (only polycubes) so for this I had to fake it by modelling the puzzle as a 2d problem on a tan-triangle grid with the four layers side by side. For each multilayered piece I had to explicitly enter all eight orientations. To eliminate the rotations of the solutions, I kept the orientation of the orange piece fixed. $\endgroup$ Jun 28, 2023 at 13:23
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    $\begingroup$ Ha! Ik had het mis. You, sir, are very clever, indeed. Thanks for the link. I've often thought about such systems, what they'd look like, since they released the eternity puzzle. I'll have a peek under the hood. :D $\endgroup$
    – J D
    Jun 28, 2023 at 16:07
  • $\begingroup$ wow thank you @JaapScherphuis! is your solver guaranteed to be exhaustive? i.e., can I assume that there are no more than the 22 you posted, especially given the clever encoding you used? $\endgroup$ Jun 30, 2023 at 6:03
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    $\begingroup$ @fromscratch It is an exhaustive search. It is based on the very reasonable assumption that the only solutions are those where all the triangular faces of the constituent prisms are all parallel. I do not believe solutions exist with non-parallel prisms, and further exhaustive searches for solutions to subsets of the finished cube seem to bear that out. $\endgroup$ Jun 30, 2023 at 7:45

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