Friday, September 19, 2014

[uewaoxcz] Coloring cube faces

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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