Parity (mathematics) - Parity For Other Objects

Parity For Other Objects

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

The two white bishops are confined to squares of opposite parity; the black knight can only jump to squares of alternating parity.

Parity is also used to refer to a number of other properties.

• The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In Rubik's Cube, Megaminx, and other twisty puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the configuration space of these puzzles.

• The parity of a function describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case f(x) = 0, to be both odd and even.

• 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 checkerboard lattice contains all integer points of even parity. This feature manifests itself in chess, as bishops are constrained to squares of the same parity; knights alternate parity between moves. This form of parity was famously used to solve the Mutilated chessboard problem.