Laws of the cube

Only a 12^th of all states reachable by assembling the cube piece-by-piece can actually be reached 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.

Recognising an Unsolvable Cube

As a consequence of the laws above, if any of the following cases appear at the end of a solve, you can be certain the cube is not solvable with legal moves:

1. Only two pieces to swap
2. A single edge to flip
3. Only a single corner to twist

This also holds true for combinations of the above. For example, everything solved apart from a single flipped edge and a single twisted corner.

See Also

• Random State Scramble

External Links

• Laws of the Cube (by Ryan Heise)