Laws of the cube

Only a 12^th of possible cube states is reachable by twisting the 6 faces (from the solved state). Here's why:

1. Only an even number of cubie swaps is possible (¹/₂ of all states)
2. Only an even number of edges can be flipped (¹/₂ of all states) ^[1]
3. The total number of corner twists must be divisible by 3 (¹/₃ of all states) ^[2]

So combined together, these laws mean that:
¹/₂ * ¹/₂ * ¹/₃ = ¹/₁₂ of all states are reachable.

Footnotes

• [1] Assuming an oriented edge is defined as one which can be solved using only <U, D, L R, F2, B2>, an edge is defined as 'flipped' if it is not oriented.
• [2] Assuming an oriented corner is defined as one which can be solved using only <U, D, R2, L2, F2, B2>, the number of corner twists can be counted as the number of counter clockwise 120° twists required to orient it.

See Also

• Random State Scramble

External Links

• Laws of the Cube (by Ryan Heise)