A Short Proof of the Nonexistence of a Pair of Orthogonal Latin Squares of Order Six

A Latin square of order s is an s by s array L of the symbols { 1,2,..., s}, such that each symbol occurs once in each row and column of L. Two Latin squares L and M of order s are orthogonal if their superposition yields all s2 possible ordered pairs (i,j), 1 < i, j < s. It was conjectured by Euler, and proved by Tarry [4] that there do not exist a pair of orthogonal Latin squares of order six (see also [2, 31). These nonexistence proofs are long and require the consideration of many cases. We give a short, self-contained noncomputer proof which requires a minimum of casework. We describe our proof in terms of transversal designs: a TD(4,6) is a triple (X, F’, s?‘), where X is a set of size 24, F is a partition of X into four subsets (groups) of X of size six, and ~8’ is a set of 36 subsets (blocks) of X, each of size four, such that any group meets any block in a point, and any two points from different groups occur in a block. It is well known that a TD(4,6) is equivalent to a pair of orthogonal Latin squares of order six. Let (X, S’, J) be a TD(4,6). Then P = (X, F U ,pP) is a PBD (pairwise balanced design) with 24 points and 40 blocks (36 of size four and four of size six). Let the points be named xi (1 < i < 24), and label the blocks Bj (1 <j< 40), where B,, B,, B,, and B, are the blocks of size six. The incidence matrix of P is the 0 1 matrix M = (mJ defined by