Laws of the cube
From Speedsolving.com Wiki
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