Making a Wooden Soma Cube Puzzle

This idea came to me from The Second Scientific American Book of Mathematical Puzzles and Diversions, a collection of puzzles presented by Matrin Gardner and first published in 1958. The puzzle is made up of 7 pieces - the first of which is composed of 3 cubes, while the remaining 6 are each composed of four cubes. Gardner provides a handy illustration of the puzzle pieces:

I used this as a guide to make my copy of the puzzle. I also needed some wooden blocks, and my local antiques store came through! I found a nice bag of wooden blocks with (mostly) uniform size and design for just \$6, tax included.

Before you say I got ripped off, this bag holds 48 red cubes, 3/4” on an edge, with numbered faces! (plus two slightly smaller blue cubes, also with numbered faces). A set of 27 3/4” wooden cubes at Lowe’s would have cost me around \$10 - and those don’t have numbered faces.

So what’s the big deal about numbered faces? The numbers give me the opportunity to add another level of curiosity to the puzzle; after all, there’s a lot you can do with numbers $^{\text{[citation needed]}}$. One thing you can do is add them together, so I thought I would try adding them to make prime numbers; I would arrange the cubes so that the sum of the visible faces on each Soma piece would be a prime number.

My first approach was to do this by hand, by drawing out a mesh for each cube (there are two kinds - one has faces numbered 0-5, and the other is numbered 5-10).

[meshes image here]

I had originally planned to only use the blocks with low-numbered faces, but I realized that if I messed up a block in the process of making the cubes, I wouldn’t have enough replacements. So I needed to balance the number of low-valued and high-valued cubes I included. At this point, the number of possible configurations for each cube was becoming overwhelming, and the brute-force approach was too taxing to undertake by hand.

I resolved this by writing a program to brute-force it for me. The code is available in this repository. The program looks at all the choices for 4 cubes, and all possible ways to hide a given number of faces (usually 2 per cube, but in piece 7, the middle cube has 3 hidden faces), and prints out the configurations for which the non-hidden faces sum to a prime number. From this output, I could freely choose configuratio ns that used a balanced number of low- and high-value cube faces.