Two Latin squares are orthogonal when their overlay shows all n² distinct pairs
Euler conjectured no orthogonal Latin squares exist for n ≡ 2 (mod 4) — n=2,6,10,... He was right for n=2 and n=6, but wrong for n≥10! The 36 Officers Problem (n=6) was proved impossible by exhaustive search (Tarry, 1901).