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Higher dimensions and more general classes of numbers

Integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the face-centered cubic lattice and its higher-dimensional that is generalizations, the D lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to moving between squares of the same parity, whereas knights alternate parity between moves. This form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other.

The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.

Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I {\displaystyle 0+I} may be called even, while elements of the coset 1 + I {\displaystyle 1+I} may be called odd.As an example, let R = Z be the localization of Z at the prime ideal (2). Then an element of R is even or odd if and only if its numerator is so in Z.