Latin square
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3x3 Latin square is:
The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3.
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 Latin square above is not reduced because its first column is A, C, B rather than A, B, C.
Any Latin square can be reduced by permuting (that is, reordering) the rows and columns. Here switching the above matrix's second and third rows yields the following square:
This Latin square is reduced; both its first row and its first column are alphabetically ordered A, B, C.
If each entry of an n × n Latin square is written as a triple (r,c,s), where r is the row, c is the column, and s is the symbol, we obtain a set of n2 triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the following Latin square is:
where for example the triple (2,3,1) means that in row 2 and column 3 there is the symbol 1. The definition of a Latin square can be written in terms of orthogonal arrays:
For any Latin square, there are n2 triples since choosing any two uniquely determines the third. (Otherwise, an ordered pair would appear more than once in the Latin square.)
The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.
Many operations on a Latin square produce another Latin square (for example, turning it upside down).
If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called isotopy classes, such that two squares in the same class are isotopic and two squares in different classes are not isotopic.
...
Wikipedia