On orthogonal Latin squares
An m-sided Latin square is an arrangement of the numbers 1, 2, , m into m rows and m columns in such a way that no row and no column contains any number twice. Two Latin squares are said to be orthogonal if when one is superimposed upon the other every ordered pair of numbers occurs once in the resulting square. Various methods have been devised for the construction of sets of orthogonal squares. However no method has as yet been given which would yield all possible sets of orthogonal Latin squares. In constructing orthogonal sets it is of value to have simple criteria which enable us to decide whether a given Latin square can be a member of an orthogonal pair.