Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively.

Let us say that I have a permutation of the stickers on the cube written as a word in the primitive moves, e.g. RUDU.

Is there a way to find the set of all equivalent words in the primitive moves (under a certain length)?

I imagine that it would be something like $\sigma H$ where $\sigma$ is the word you start with and $H$ is a subgroup of $\langle R,L,U,D,F,B\rangle$.

Unfortunately, it's been awhile since I've done anything with group theory, so any help would be much appreciated.


Any algorithm for this will be no more efficient than an algorithm for solving the cube in a minimal number of moves. If you had an efficient algorithm for doing this you could use it to solve an arbitrarily mixed up cube. Just feed the algorithm the moves used to scramble the cube and use binary search to find the smallest length series of moves equivalent to the scramble sequence.


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