The "25Y" or "125 cube" puzzle 



When my program started working and listed several solutions to the problem I observed that some solutions differed only by the position of two puzzle pieces. At first I thought that there was still something wrong with the program because I could not imagine how this could be possible. But a closer examination revealed that in these cases just the "spike" of a piece  the cube which is not collinear with the other 4 cubes  exchanged between two adjacent pieces. This can only happen in a few different configurations:
Let us now call a solution to the puzzle isolated, if it cannot be derived from another solution by a spike swap. Obviously, this is the case if and only if adjacent pieces in the puzzle never are in a configuration depicted above. 

I wondered how many of the 1264 solutions are isolated solutions. The computer analysis with the SATsolver showed in about 1 hour CPUtime that there are just 5 isolated solutions. They are depicted here. 

[1] Chris Bouwkamp, The CubeY Problem, Cubism For Fun 25 (December 1990  January 1991), part 3, pp. 3043. (includes a listing of all 1264 solutions) 

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