ABSTRACT
A Latin square of side (or order) n is an n×n array based on some set S of n symbols (treatments), with the property that every row and every column contains every symbol exactly once. In other words, every row and every column is a permutation of S. Since the arithmetical properties of the symbols are not used, the nature of the elements of S is immaterial; unless otherwise specified, we take them to be {1, 2, . . . , n}. As an example, let G be a finite group with n elements g1, g2, . . . , gn. Define an array A = (aij) by aij = k where k is the integer such that gk = gigj (the multiplication table of G). We look at column j of A. Consider all the elements gigj, for different elements gi. Recall that gj has an inverse element in G, written g−1j . If i and k are any two integers from 1 to n, then gigj = gkgj would imply gigjg−1j = gkgjg
whence gi = gk, so i = k. So the elements g1gj , g2gj, . . . , gngj must all be different. This means that the (1, j), (2, j), . . . , (n, j) entries of the array are all different, so column j contains a permutation of {1, 2, . . . , n}. This is true of every column. A similar proof applies to rows. So A is a Latin square. This example gives rise to infinitely many Latin squares. For example, at sides 1, 2 and 3 we have
1 1 22 1
1 2 3 2 3 1 3 1 2
There are Latin squares of side 3 that look quite different from the one just given, for example
1 2 3 3 1 2 2 3 1
but this can be converted into the given 3 × 3 square by exchanging rows 2 and 3.
