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Latin Squares in Practice and in Theory II

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2. Orthogonal Latin Squares

A latin square of size n is an array of n copies each of n different objects (typically the latin letters A, B, C, ...) so that all the objects in any row, and all the objects in any column, are different. Two size n latin squares, one with objects A, B, C, ..., one with objects a, b, c, are orthogonal if superimposing them leads to a square array containing all n2 possibile pairs (A,a), (A,b), ... , (B,a), (B,b), ..., ... .

For example, the two 5 x 5 latin squares

are orthogonal: they can be superimposed to give every possible combination of rank and color

If we could find two orthogonal latin squares of size 6, they would combine to give a solution to Euler's problem of the 36 officers. So an equivalent statement to the impossibility of solving that problem is: There are no two orthogonal latin squares of size 6.


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