nLab Latin square

An $n \times n$ Latin square is a square array of numbers from $1$ to $n$ such that each row and each column contains every number from $1$ to $n$. For example, an ordinary sudoku square is a special type of $9 \times 9$ Latin square.

Latin squares can be regarded as precisely the multiplication tables for quasigroup structures on the set $\{1, 2, \ldots, n\}$, where $i j$ is the entry in the $i^{th}$ row and $j^{th}$ column. Indeed, the condition that no two entries in the $i^{th}$ row are the same says that left multiplication by $i$ is invertible, and that no two entries in the $j^{th}$ column are the same says that right multiplication by $j$ is invertible.

category: combinatorics