Parity in a puzzle describes whether a move or sequence of moves swaps pieces an even or odd amount of times. If a puzzle is described as having Even Parity, only moves with even-parity are possible. If a puzzle is described as having Odd Parity then it can perform both Odd and Even parity moves. For example, a standard 3-cycle (ABC -> BCA) would have Even parity because it is actually 2 swaps A<->B and A<->C. Orientation of pieces is ignored when considering swapped pieces.
The term Parity is often misused in cubing. It is used colloquially to describe an odd permutation of pieces within a certain defined orbital. It is most commonly used to refer to certain situations that can arise on the 4x4x4 and 5x5x5 cubes when solving with the reduction method. The most commonly known example is referred to as OLL Parity, and comes about on cubes of size 4x4x4 and larger. OLL Parity on a 4x4x4 cube happens when two adjacent wing edges are interchanged with each other. When solving with the reduction method these two interchanged wing edges give the two piece edge group the appearance of being a flipped 3x3x3 edge. This leads to a case that is impossible to solve using outer layer turns only, and must be solved by performing an odd number of inner slice quarter turns to return the wing edge orbital back to an even permutation.
- Speedsolving.com: Understanding Parity
- Parity problem - detailed explanation of the term "parity problem" and its application on the 4x4x4 Supercube, even and odd permutation