# 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:

**Only an even number of cubie swaps is possible**(^{1}/_{2}of all permutations)^{[1]}**Only an even number of edges can be in a flipped state**(^{1}/_{2}of all edge orientations)^{[2]}**The total number of corner twists must be divisible by 3**(^{1}/_{3}of all corner orientations)^{[3]}

So combined together, these laws mean that:

^{1}/_{2} * ^{1}/_{2} * ^{1}/_{3} = ^{1}/_{12} of all states are reachable.

**Footnotes**

- [1] Assuming permutations are hypothetically solved using a sequence cubie swaps.
- [2] An edge is defined as 'flipped' if cannot be solved using only <U, D, L R, F2, B2>.
- [3] If 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:

**Only two pieces to swap****Only a single edge to flip****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

## External links

- Laws of the Cube (by Ryan Heise)
- Wikipedia: Parity of a Permutation
- Speedsolving.com: The laws of a Rubik's Cube
- Speedsolving.com: Unreachable positions