Rubiks Cube/Numbers

Number of permutations of 3x3

To count number of permutations of 3x3, count permutations due to both the placement and orientation of each type of piece (corners and edges; centers are fixed):

• 3 possible rotations of 8 corners, 7 corners determine the 8th = $ 3^7 $
• 2 possible rotations of 12 edges, but only 11 edges can be flipped independently; the orientation of the 12th edge is forced by the other 11 = $ 2^{11} $
• 8 different corner pieces to be distributed to 8 locations = $ 8! $
• 12 different edge pieces to be distributed to 12 locations = $ 12! $
• Half of the cube permutations (those requiring an odd permutation of edges/corners) cannot be reached without disassembling the cube = $ \frac{1}{2} $

Correction (April 2025): The edge-orientation factor was previously listed as $ 2^{12} $. Only 11 of the 12 edges can be flipped independently; the final edge's orientation is determined by the parity of the other 11. This constraint is separate from the $ \tfrac{1}{2} $ factor already applied for the even-permutation requirement. Fixing this brings the count from ~86 quintillion down to the correct value of ~43 quintillion, matching the widely published number (e.g., Wikipedia). The 5×5 section below already correctly notes this distinction.

Total number of permutations of a 3x3 cube:

$ N = 3^{7} \times 8! \times 2^{11} \times 12! \times \frac{1}{2} = 43252003274489856000 \sim 4.32520 \times 10^{19} $

or about 43 quintillion permutations.

Counting Cycles

Length of Cycles on 3x3

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.