Graeco-Latin square

A Graeco-Latin square or Euler square 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 ∈ S and a t ∈ T, such that

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every row and every column contains exactly one s ∈ S and exactly one t ∈ T, and

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no two cells contain the same ordered pair of symbols.

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The two sets are commonly taken to be S = {A, B, C, …}, the first n upper-case letters from the Latin alphabet, and T = {α , β, γ, …},

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the first n lower-case letters from the Greek alphabet—hence the name Graeco-Latin square. Several examples are given below.

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Order 3

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Order 4

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Order 5

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The arrangement of the Latin characters alone and of the Greek characters alone 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.

Related Topics:
Latin square - Orthogonal - Cartesian product

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Graeco-Latin squares have applications in the design of experiments, and can be used in the construction of multiplicative magic squares.

Related Topics:
Design of experiments - Magic square

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