Mutually orthogonal Latin squares

A collection of Latin squares of the same order in which every pair is orthogonal: superimposing any two produces each ordered pair of symbols exactly once.

Latin square: An array filled with symbols so that each symbol appears exactly once in every row and column. Each member of the collection must satisfy this row-and-column condition.
Permutation: A rearrangement of a finite set that assigns each element to exactly one position. Every row and column of a Latin square is a permutation of its symbols.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped number theory, analysis, and combinatorics. Euler studied orthogonal Latin squares and posed the famous order-six problem.
Design of experiments: The planning of controlled studies to estimate effects and interactions efficiently and reliably. Orthogonal arrangements help balance treatment combinations across experimental factors.
Euler square: A generalization of a Latin square in which symbols are arranged to satisfy specified row, column, and subsquare conditions. The name is sometimes confused with orthogonal Latin squares, but the defining constraints differ.
Orthogonal Latin squares: Two Latin squares of the same order whose superposition contains every ordered pair of symbols exactly once. Mutual orthogonality requires this relation for every pair in the collection.
Complete set of mutually orthogonal Latin squares: A family of order n containing the maximum possible n−1 mutually orthogonal Latin squares. It represents the largest family attainable for orders where that bound is reached.
Euler's officers problem: A puzzle asking whether 36 officers from six ranks and six regiments can be arranged in a square so each rank and regiment occurs once per row and column. Its arrangement is equivalent to a pair of orthogonal Latin squares of order six.
Graeco-Latin square: A pair of orthogonal Latin squares whose two symbol sets are often represented by Latin and Greek letters. It is a direct experimental layout controlling two blocking factors alongside treatments.
Mutually orthogonal diagonal Latin squares: A family of pairwise orthogonal Latin squares that also satisfy diagonal symbol conditions. The added diagonal requirements make these a stricter variant of the present concept.

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