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: