Latin square

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In combinatorics and in experimental design, a Latin square is an n × n array filled with n different Latin letters, each occurring exactly once in each row and exactly once in each column. Here is an example:

The name "Latin square" is motivated by mathematical papers by Leonhard Euler, who used Latin characters as symbols. Of course, other symbols can be used instead of Latin letters: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3.

Contents

Overview of uses

Latin squares are used in statistics and in mathematics.

Reduced form

A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. For example, the above Latin square is not reduced because its first column is A, C, B rather than A, B, C.

We can make any Latin square reduced by permuting (reordering) the rows and columns. Here switching the above matrix's second and third rows yields

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