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# Encyclopaedia of DesignTheory: Latin squares

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Topic essays:

# Latin squares

A Latin square of order n is an n x n array, in which each cell of the array contains one of a set of n symbols, in such a way that each symbol occurs once in each row and once in each column.

Here is an example.

 1 2 3 2 3 1 3 1 2

There is no need for the symbols to be the numbers 1,...,n: any symbols will do, even colours:

Latin squares of all possible orders exist. The cyclic structure of the example above generalises immediately to any size. More generally, the Cayley table of any group is a Latin square; the above examples come from the cyclic group of order 3.

Latin squares are "equivalent" to many other types of combinatorial structures. An account of some of these, and the varying notions of isomorphism to which they give rise, is given in a topic essay.

Peter J. Cameron
4 October 2004