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Hoey showed using the Cauchy-Frobenius Lemma that there are positions up to conjugacy by whole-cube symmetries. "Metamagical Themas: The Magic Cube's Cubies are Twiddled by Cubists and Solved by Cubemeisters." Sci.

The group of operations on Rubik's Cube is known as Rubik's group, and the Cayley graph of that group is called Rubik's graph.

14 in Metamagical Themas: Questing of Mind and Pattern.

"Rubik's Revenge: The Group Theoretical Solution." Amer.

The minimum number of turns required to solve the cube from an arbitrary starting position is equal to the graph diameter of Rubik's graph, and is sometimes known as God's number. Schoenert/Cube-Lovers/Dan_Hoey__The_real_size_of_cube_

While algorithms exist for solving a cube from an arbitrary initial position, they are not necessarily optimal (i.e., requiring a minimum number of turns) and computation of God's number is very difficult.

Solving the cube blindfolded requires considerably less algorithms than solving it sighted.The puzzle was invented in the 1970s by the Hungarian Ernő Rubik and sold millions of copies worldwide over the next decade. The number of possible positions of Rubik's Cube is (Turner and Gold 1985, Schönert). It had been known since 1995 that a lower bound on the number of moves for the solution (in the worst case) was 20, it was not known until demonstrated by Rokicki et al. (2010) that no configuration requires more than 20 moves, thus establishing that God's number is 20.

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