How many ways are there to color the faces of a cube? Two colorings are considered equivalent if they can be transformed to each other by rotation, reflection, or color permutation.
1 color, trivially 1 way.
2 colors. Ratio 1:5 = 1 way, ratio 2:4 = 2 ways, ratio 3:3 = 2 ways. Total 5.
3 colors. Ratio 1:1:4 = 2 ways, ratio 1:2:3 = 3 ways, ratio 2:2:2 = 3 ways. Total 8.
4 colors. Ratio 1:1:1:3 = 2 ways, ratio 1:1:2:2 = 3+1=4 ways. Total 6.
5 colors. Ratio 1:1:1:1:2 = 2 ways.
6 colors. 1 way.
Grand total 23.
The principled method would have been the Burnside lemma, article by Nick Baxter, which could tackle other polyhedra. http://baxterweb.com/puzzles/burnside5.pdf
23 is not so different from the letters of the alphabet, suggesting the invention of a very 3D alphabet: no 2D snapshot is sufficient.
Inspired by considering easier Rubik's cubes.
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