Definition:Latin Square

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 $4$:

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

Historical Note

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