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Encyclopaedia of DesignTheory: Latin squares

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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.

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Peter J. Cameron
16 April 2002