Laws of the cube

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Only a 12th 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 permutations) [1]
  2. Only an even number of edges can be in a flipped state (1/2 of all edge orientations) [2]
  3. 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:


  1. Only two pieces to swap
  2. Only 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

External links