Quantum solutions to Euler’s 36 officers

Yesterday Simeon Ball sent me notices, in New Scientist and phys.org, of his paper with Robin Simoens on quantum solutions to Euler’s 36 officers problem: Given 36 officers belonging to six different ranks and six different regiments, each rank-regiment combination represented by one officer, arrange them on a parade ground inn a 6×6 array so that each row and column of the array contains one officer of each rank and one from each regiment. In modern terminology, this asks for a pair of orthogonal Latin squares of order 6, As is well known, Euler tried to solve the problem and failed, though he was able to establish the existence of two orthogonal Latin squares of every order not congrueent to 2 (mod 4). He conjectured that, for orders congruent to 2 (mod 4), no such squares could exist. It was more than 100 years later that Tarry showed, by exhaustive search, that Euler was right about 6; no pair of squares exists. More than half a century after that, Bose, Shrikhande and Parker…