Latin square

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Latin square is a concept in combinatorics, a branch of mathematics that deals with the arrangement, combination, and permutation of sets. A Latin square of order n is an n×n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. The name "Latin square" was inspired by Leonhard Euler, who used Latin characters as symbols.

Definition

A Latin square of order n is defined as an n×n grid in which each cell contains a single symbol from a set of n symbols, satisfying the condition that each symbol appears exactly once in each row and exactly once in each column. For example, a Latin square of order 3 might use the symbols 1, 2, and 3, and be arranged as follows:

\[ \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \\ \end{array} \]

Properties

Latin squares exhibit several interesting properties:

• Two Latin squares are said to be orthogonal if, when superimposed, each ordered pair of symbols occurs exactly once. A set of Latin squares in which each pair is orthogonal is known as a set of mutually orthogonal Latin squares (MOLS).
• A Latin square that remains unchanged when rotated 180 degrees is called a symmetric Latin square.
• The concept of a reduced or normalized Latin square refers to a Latin square where the first row and the first column are in natural order of the symbols.

Applications

Latin squares have applications in various fields including experimental design, error correcting codes, and puzzle creation. In experimental design, Latin squares are used to control for two nuisance variables, thereby isolating the effect of the primary variable of interest. They are also foundational in the construction of certain types of error correcting codes and are the basis for popular puzzles like Sudoku.

Related Concepts

• Sudoku: A puzzle that is essentially a partially completed 9×9 Latin square, where the challenge is to fill in the missing numbers.
• Orthogonal Array: A generalization of Latin squares that allows for more than two dimensions.
• Magic Square: Although not the same, magic squares are related in that they are square grids with numbers that meet certain sums, but they do not have the requirement of each symbol appearing exactly once per row and column.

See Also

• Combinatorial Design
• Graph Theory
• Permutation