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
Order $3$ Latin Square
This is an example of a Latin square of order $3$:
$\quad \begin {array} {|ccc|} \hline A & B & C \\ C & A & B \\ B & C & A \\ \hline \end {array}$
Order $4$ Latin Square
This is an example of a Latin square of order $4$:
$\quad \begin {array} {|cccc|} \hline a & b & c & d \\ c & d & a & b \\ d & c & b & a \\ b & a & d & c \\ \hline \end {array}$
In Context of Experimental Design
In the context of design theory, a Latin square allows classification by three mutually orthogonal factors which can be denoted by rows, columns and conventional "Latin" letters.
Treatments are designated by letters, and allocated to units under restricted randomization, each treatment occurring exactly once in each row and column.
Hence the Latin square provides a useful double-blocking system to increase precision, by reducing two potential sources of variation not relating to treatments.
In the analysis of variance, the degrees of freedom for the error mean square are low for Latin squares smaller than order $6$, but this problem can be overcome by using more than one Latin square.
The fact that the number of treatments must equal the number of rows or columns can lead to difficulties in practice.
Also see
- Existence of Latin Squares: Latin squares exist for all $n$.
- Results about Latin squares can be found here.
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): Chapter $\text {I}$: The Group Concept: $\S 5$: The Multiplication Table
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Latin square
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Latin square
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Latin square