Post #4327327
2026-08-01 07:30 UTC
@OscarCunningham@mathstodon.xyz i wrote a program that enumerates all the colorings, and it doesn't find any that work: http://ianhenderson.org/2026-08-01-dodecahedron.c.txt (the face numberings were from putting labels on a real-life icosahedron i had handy, so i could have made mistakes during that process, but the group it generates does have order 60 at least)
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@OscarCunningham@mathstodon.xyz 2026-08-01 08:07
@ianh@mastodon.social Nice! I was coming to the same conclusion myself. I worked out that there are only 12 possible arrangements of black/white around a face that don't have rotation symmetry. So we would need to use each one exactly once. In particular there will be a white face with only one black face touching it, and a black face with only one white face touching it. These sets of six faces can only overlap in a small number of ways. Then you can check that you can't fill in the remaining faces in any way that works.