# Definition:Latin Square

## Contents

## Definition

Let $n \in \Z_{>0}$ be some given (strictly) positive integer $n$.

A **Latin square** of order $n$ is a square array of size $n \times n$ containing $n$ different symbols, such that every row and column contains **exactly one** of each symbol.

That is, each row and column is a permutation of the same $n$ symbols.

### Order of Latin Square

Let $\mathbf L$ be an $n \times n$ Latin square.

The **order** of $\mathbf L$ is $n$.

### Row of Latin Square

Let $\mathbf L$ be a Latin square.

The **rows** of $\mathbf L$ are the lines of elements reading **across** the page.

### Column of Latin Square

Let $\mathbf L$ be a Latin square.

The **columns** of $\mathbf L$ are the lines of elements reading **down** the page.

### Element of Latin Square

Let $\mathbf L$ be a Latin square of order $n$.

The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the **elements** of $\mathbf L$.

The element at row $i$ and column $j$ is called **element $\left({i, j}\right)$ of $\mathbf L$**, and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character.

If the indices are still more complicated coefficients and further clarity is required, then the form $a \left({i, j}\right)$ can be used.

Note that the first subscript determines the row, and the second the column, of the Latin square where the element is positioned.

## Examples

This is an example of a **Latin square of order $4$**:

- $\begin{array} {|cccc|} \hline a & b & c & d \\ c & d & a & b \\ d & c & b & a \\ b & a & d & c \\ \hline \end{array}$

## Also see

- Existence of Latin Squares: Latin squares exist for all $n$.

## Historical Note

The concept of a **Latin square** originates from Leonhard Paul Euler, who used Latin characters as symbols.

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 5$: The Multiplication Table