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Latin squares as arrays

There is another important representation of a Latin square as an array. Assume that the symbols in the square are 1,2,...,n. Then let S be the set of n² triples of the form (i,j,k), where the symbol in row i and column j of the square is k.

For example, given the square

1	2	3
2	3	1
3	1	2

we obtain the following nine triples:

1	1	1
1	2	2
1	3	3
2	1	2
2	2	3
2	3	1
3	1	3
3	2	1
3	3	2

This is an orthogonal array of strength 2 and index 1. This means that, if you slide your fingers down any two columns of the array, you will see each ordered pair of symbols precisely once.

Conversely, from any orthogonal array of strength 2 and index 1 with 3 columns, we can reconstruct a Latin square, by putting symbol k in row i and column j if the row (i,j,k) occurs in the array.

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Peter J. Cameron
23 October 2002