# The "25Y" or "125 cube" puzzle

 The puzzle is made of 13 of the Y-shaped pentomino consisting of 3 white cubes and 2 dark cubes and 12 Y-pentominoes of 2 withe cubes and 3 dark cubes. For this reason, the puzzle is called 25Y-puzzle sometimes. The goal is to pack these pieces into a 5x5x5 box generating a checkerboard pattern. The task of packing the pieces into the box is the really demanding part - if you have managed this the creation of the checkerboard is almost trivial and we will ignore this from here on. Since I have never managed to find a solution by hand I wrote a program which is able to solve the problem.by transfroming it into a boolean satisfiability problem and let a SAT-solver do the rest of the job. It is well known that there are 1264 possible ways to pack the box if the solutions are reduced by the 48 symmetries of the cube [1].

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 SAT-solver showed in about 1 hour CPU-time that there are just 5 isolated solutions. They are depicted here.

[1] Chris Bouwkamp, The Cube-Y Problem, Cubism For Fun 25 (December 1990 - January 1991), part 3, pp. 30-43. (includes a listing of all 1264 solutions)
[2] C.J. Bouwkamp and D.A. Klarner, Packing a Box with Y-pentacubes, Journal of Recreational Mathematics 3 (1970), no. 1, pp. 10-26.