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Concept
Mutually orthogonal Latin squares
Summary
In combinatorial mathematics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.
An outdated term for pair of orthogonal Latin squares is Graeco-Latin square, found in older literature.
A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), 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.
GraecoLatinSquare-Order3.svg|Order 3
GrecoLatinSquare-Order4.svg|Order 4
GraecoLatinSquare-Order5.png|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 were studied in detail by Leonhard Euler, who took the two sets to be S = {A, B, C, ...}, the first n upper-case letters from the Latin alphabet, and T = {α , β, γ, ...},
the first n lower-case letters from the Greek alphabet—hence the name Graeco-Latin square.
When a Graeco-Latin square is viewed as a pair of orthogonal Latin squares, each of the Latin squares is said to have an orthogonal mate.
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