What percentage of the total number of combinations on a Rubik's Cube have the following property:
On every face, no two adjacent squares have the same color.
By adjacent I mean:
adjacent: sharing a common side.
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Truth-seek
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I have no idea of the exact answer. There are 72 pairs of adjacent squares, so perhaps it would be near $(5/6)^{72}$ or around 2 in a million.
Empy2
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It can't be that low. I almost randomly hit on a combination just like that, after making something like 10 or 15 moves. It just can't be 2 in a million – Truth-seek Feb 18 '18 at 16:53
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Another argument: Each edge may be in 17 of the 24 positions to avoid matching a centre cubie. $(17/24)^{12}=0.016$. The chance the eight corners in any particular arrangement cause no clash would be $(4/6)^{24}$, or maybe $(4/6)^{20}(5/6)^4=0.000145$. The product is 2.3 millionths. – Empy2 Feb 19 '18 at 10:30
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@Truth-seek were your moves uniformly random? I'm guessing maybe you preferred moves that broke up adjacent colors (as I would do if trying to intentionally scramble the puzzle). In that case you'd naturally get to a configuration with the given property pretty quickly; that doesn't sound incompatible with there being 2 in a million of them. – Don Hatch Sep 24 '18 at 08:27