Mutually orthogonal Latin squares

Summary

Two Latin squares of the same order are orthogonal if, when one is laid over the other, every ordered pair of symbols appears exactly once. A family of Latin squares is mutually orthogonal if every two of them are orthogonal. For $n \ge 2$, there are at most $n - 1$ mutually orthogonal Latin squares of order $n$. A family of $n - 1$ of them exists exactly when a finite projective plane of order $n$ exists. The node mutually orthogonal Latin squares gives the precise definition.