Mutually orthogonal latin square

In combinatorics, a Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n×n arrangement of cells, each cell containing an ordered pair (s,t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.

Order 3

Order 4

Order 5

The arrangement of the s-coordinates by themselves (which may be thought of as Latin characters) and of the t-coordinates (the Greek characters) each forms a Latin square. A Graeco-Latin square can therefore be decomposed into two "orthogonal" Latin squares. Orthogonality here means that every pair (s, t) from the Cartesian product S×T occurs exactly once.

Orthogonal Latin squares have been known to predate Euler. As described by Donald Knuth in Volume 4A, p. 3, of TAOCP, the construction of 4x4 set was published by Jacques Ozanam in 1725 (in Recreation mathematiques et physiques, Vol. IV) as a puzzle involving playing cards. The problem was to take all aces, kings, queens and jacks from a standard deck of cards, and arrange them in a 4x4 grid such that each row and each column contained all four suits as well as one of each face value. This problem has several solutions.

A common variant of this problem was to arrange the 16 cards so that, in addition to the row and column constraints, each diagonal contains all four face values and all four suits as well.

According to Martin Gardner, who featured this problem in his November 1959 Mathematical Games column, the number of distinct solutions was incorrectly stated to be 72 by Rouse Ball. This mistake persisted for many years until the correct value of 144 was found by Kathleen Ollerenshaw. Each of the 144 solutions has eight reflections and rotations, giving 1152 solutions in total. The 144×8 solutions can be categorized into the following two equivalence classes:

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