A while back I supported a Kickstarter campaign for custom laser-engraved dice. I figured there had to be lots of mathy designs out there waiting to be found, and being able to make physical copies of them would inspire me to find some of them. And I have.
One of the first ideas i had was to use magic cubes. I didn’t know what sorts of magic cubes had been discovered, but I knew it was an obvious enough variation on the idea of magic squares that something had to be out there. In general, most magic cubes aren’t good for printing on dice, because they contain numbers in the interior of the cube, and one typically is only able to engrave the exterior. I did however find a couple of promising variations on the page on Unusual Magic Cubes at the late Harvey Heinz’s magic-squares.net site.
That page shows a cube found by Mirko Dobnik where each face is divided into a 2×2 grid. The faces sum to 50, and the rings around the cube sum to 100. In order to turn Dobnik’s cube into a usable six-sided die, I would need to find a solution that placed the numbers 1 to 6 on different faces of the cube, ideally on the same faces that they would occupy on a standard die. Then I could set these numbers in boldface to show that they represent the result of a die roll for a given side.
I decided this was a good time to learn how to use a constraint solver. I picked Numberjack because it uses Python, which is the language I am most comfortable with, and there was a magic square example that I could tweak. With face sum and ring sum constraints, and constraints to put the numbers 1 to 6 on the proper faces, (plus symmetry breaking constraints) I was getting at least hundreds of thousands of solutions. So I added constraints to make the four diagonals that traverse all six faces to sum to 75, and I fixed the positions of the numbers 1-6. The most symmetrical ways to pick corners of the faces of a die are to take all of the ones on one diagonal ring, or to take the corners that meet at antipodal vertices of the die. The first would violate the diagonal sum constraint, and the second bunched the relevant numbers up more than I liked, so I picked an arrangement that still has rotational symmetry about one of the axes through antipodal vertices, but that doesn’t have reflection symmetry. Then there are four ways to pick the axis of symmetry. At first I chose one at random, and came up with just eight solutions, one of which had odds and evens in a checkered pattern. Then I tried again with the axis going through the [1,2,3] and [4,5,6] vertices, and there were two parity-checkered solutions out of six, one of which is shown above.
I know it’s a bit strange to disappear from blogging for nearly a year, and then promise a multipart post that is itself part of a series, but yes, that is just what I’m doing. There are more magic dice to come, and then more mathematically inspired dice that are less magic. Hopefully there will also be some posts that are not about dice, not too far off.