For which n is there a pair of orthogonal Latin squares?

n odd: YES (problem sheet 3)

n = 2^{r} for r ≥ 2, i.e., n = 4, 8, 16, 32, …: YES (finite field theory)

n = n_{1}n_{2} where it’s true for n_{1} and n_{2}: YES

It follows that the answer is yes except perhaps when n ≡ 2 (mod 4), i.e., n = 2, 6, 10, 14 . . ..

n = 2: NO (trivial)

n = 6: NO (Euler’s officers problem) Euler conjectured in 1779 that there was no solution. Proved by Tarry in 1900.

n = 10, 14, 18, ...: YES Euler conjectured that there are no examples. Bose, Parker and Shrikhande (1960) constructed examples for all such n.