Combinations of a Rubik's Cube

Question

I recently got a Rubik's Cube and learnt how to solve it, and as anyone has I tried shuffling it to the 'hardest' possible state to solve from, for which my criteria was: none of the same coloured sides must be touching linearly (I now know the hardest state is the 'superflip' but this question is still interesting).

I know of one state which is done by: R2 L2 U2 D2 F2 B2. This creates a nice pattern.

How many other possible permutations are possible where none of the same coloured squares are touching? (And is this relatively high or low compared to the total number of permutations?)

Answer 1

The exact number will be very hard to find, but I can try to calculate a ballpark figure by making some simplifying assumptions.

The simplest is to assume that each adjacency is independent of any other, and has a probability of $1/6$ of having matching colours. There are $6\cdot12=72$ adjacent pairs of stickers, so there is a probability of about $(5/6)^{72}\approx 0.000002$ of having no matching colours anywhere. In other words, out of every million positions about $2$ have no matching colours.

We can do a slightly better approximation by looking at pairs of adjacencies. The probability of an edge not matching either of the adjacent centres is not $(5/6)^2$, but $17/24$ because there are $24$ ways to place the edge and $7$ of them have at least one matching centre. Similarly, of the $24$ ways to place a corner next to a particular side of an edge, $17$ have no matchings. This leads to a probability of $(17/24)^{36}$ for the whole cube to have no matchings, or about $4$ in a million.

Of course this still rests on the assumption that these $36$ pairs of adjacencies are independent of each other, so the actual value is slightly higher still, but it is in the right ballpark.