Mutually orthogonal Latin squares

Latin squares on the same array are mutually orthogonal when every ordered pair of symbols from any two squares occurs exactly once on superposition. Pairwise balance makes the centered symbol-indicator spaces orthogonal in an additive linear regression model.

A Graeco-Latin square is the superposition of two orthogonal Latin squares of the same order. Each symbol of either square occurs once per row and column, and every cross-square pair occurs once. It permits four mutually orthogonal additive factor spaces: rows, columns, and the two symbol factors.