Rubiks Cube/Numbers

Number of permutations

• 3 possible rotations of 8 corners, 7 corners determine the 8th = 3^7
• 2 possible rotations of 12 edges, 11 edges determine the 12th = 2^11
• 8 different corner pieces to be distributed to 8 locations = 8!
• 12 different edge pieces to be distributed to 12 locations = 12!

Furthermore, some of these possibilities are not possible. Only even parity cases can occur on the 3x3 cube, cutting the number of ways of distributing the corners and edges in half.

Total number of permutations of a 3x3 cube:

$ N = \dfrac{ 3^{7} \times 2^{11} \times 8! \times 12! }{ 2 } = 43,252,003,274,489,856,000 $

or about 43 quintillion permutations.

Length of Cycles

One of the interesting features of a Rubik's Cube is that, starting from a solved cube, if any sequence of moves is repeated long enough on the cube, will eventually result in the cube returning to the solved state.

The simplest example is a single move (like U): after executing U four times, the cube is returned to the solved state.

It is a little less trivial with a sequence of moves like U'R'UR or LU, but these also eventually result in returning the cube to its solved state after 6 repetitions of the sequence U'R'UR or 105 repetitions of the LU sequence.