# Permutation

**Permutation** is a mathematical term that refers broadly to rearranging or the rearrangement of objects. In puzzle theory, the term is applied to puzzle pieces. The permutation of pieces refers to their location on the puzzle or, equivalently, to the relocation required to place them in their correct location. One also speaks of a single piece's permutation to mean its location or where it needs to be moved, which contrasts with its orientation, its state (flip/orientation) in place.

## Relation to orientation

Although (changes in) permutation and orientation suffice to characterize any legal move on many puzzles, they are in general not independent. In mathematical terms, the group of the puzzle cannot in general be decomposed as a *direct* product of the permutation group and the orientation group. For example, on the 3x3, the decomposition is instead a *semi-direct* product; a general permutation affects orientation.

Orientation and permutation may be viewed independently by restricting to *pure orientation* or *pure permutation*, affecting one but not the other. For example, in 3OP blindfold solving method, pieces are first oriented in place (trivial permutation), then permuted while preserving the orientation. Here, the precise definition of orientation affects which permutations preserve orientation, hence affect the notion of pure permutation.

## Permutation parity

On a puzzle with different types of pieces, the permutation naturally breaks down into permutations of the different piece types. For example, the full permutation of a 3x3 can be specified by the permutation of its 12 edges and the permutation of its 8 corners. On many puzzles, however, group theoretic considerations show that there are constraints on the possible permutations of different piece types. For example, every legal move on a 3x3 has an even full permutation; corner and edge permutations must be either even and even or odd and odd. This means in particular that a single pair of edges cannot be switched without also switching a single pair of corners.

A **permutation parity** refers to a situation where the full permutation cannot be corrected using only permutations of a single piece type. On a 3x3, this means odd corner permutation and odd edge permutation, so that pair of corners and a pair of edges must be switched to *fix the parity.* This makes it impossible to treat the corners and edges completely independently, which has unpleasant consequences in 3x3 blindfold cubing. In cubes bigger than 3x3, the center pieces of the same type and of the same color are not distinguishable, which makes it impossible to determine the parity of their permutation from centers alone. In 4x4 (or any even number) reduction methods, which solve the center first, this can lead a full odd permutation in the 3x3 phase that cannot be resolved without permuting the already "solved" centers, another example of a permutation parity.