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How many ways can cube sides be colored by 4 types of color? I need something, not exactly the answer, maybe just formulas or theory I need. Up to 4 different colors. no, rotation does not change.

Vlad Trotsenko
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  • All $4$ colours or up to $4$? Do rotations of the cube which look the same count as different? – Henry May 10 '18 at 07:34
  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos May 10 '18 at 07:36
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    See https://math.stackexchange.com/questions/64857/painting-the-faces-of-a-cube-with-distinct-colours?rq=1 – Robert Z May 10 '18 at 07:36
  • up to 4 different colors. no, rotation does not change the cube colors – Vlad Trotsenko May 10 '18 at 07:36
  • Ive seen the topic you linked, but my problem is more complex since my cube can be colored using only one color – Vlad Trotsenko May 10 '18 at 07:38
  • @VladTrotsenko so repeat the process from the link 4 times: for 1 color, for 2 colors, for 3 colors and for 4 colors – Holo May 10 '18 at 07:45
  • @Holo actually it wont work since cube cannot be colored by 4 different colors because there are not enough sides! The question linked has a formula for number of colors higher than 5 because another way this question would be impossible to answer – Vlad Trotsenko May 10 '18 at 07:55

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This is a case for Burnside's Lemma, AKA Polya counting theory. The Wikipedia article explains exactly your problem in detail, albeit with three colors instead of four. Now translate the described procedure to your situation.

Christian Blatter
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