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Rubik's Cube theory

Parity

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The parity of a permutation refers to whether that permutation is even or odd. An even permutation is one that can be represented by an even number of swaps while an odd permutation is one that can be represented by an odd number of swaps.

When considering the permutation of all edges and corners together, the overall parity must be even, as dictated by laws of the cube. However, when considering only edges or corners alone, it is possible for their parity to be either even or odd. To obey the laws of the cube, if the edge parity is even then the corner parity must also be even, and if the edge parity is odd then the corner parity must also be odd.

An interesting fact is that a commutator always represents an even permutation on both edges and corners. Given a commutator X.Y.X-1.Y-1, regardless of whether X.Y involves an even or odd number of swaps, X-1.Y-1 will involve the exact same number of swaps, and 2 times any number gives us an even number overall. What this means is that commutators cannot directly solve positions where the edges and corners have odd parity.

While parity tends not to be an issue at the beginning of a solve, it may become an issue in the endgame. If the edges and corners have odd parity, the easiest way to correct them is to apply a single 90 degree turn of any face. However, in the endgame, we typically have a need to employ sequences that affect only a select few pieces while preserving everything else, and a single 90 degree turn does quite the opposite, dislodging 8 pieces. Commutators allow us to affect a small number of pieces while preserving the rest, however these do not directly apply when the parity of the edges and corners are odd.

In the Heise method, parity is dealt with while solving the last 5 edges. If all edges are solved (which is an even edge permutation), this forces the remaining corners to have even parity.

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