Latin square

Remarks

Example: for $i, j \in \set{0, 1, \dots, n - 1}$, put $L_{ij} = (i + j) \bmod n$. In row $i$ the map $j \mapsto i + j \bmod n$ is a bijection of $\set{0, \dots, n-1}$, so each symbol occurs once; the same holds for each column. This square is the addition table of the integers modulo n. More generally, the Cayley table of each finite group is a Latin square. Non-example: the array with rows $1\,2$ and $1\,2$ is not a Latin square, because the symbol $1$ occurs twice in the first column. A Latin square is the same thing as the multiplication table of a finite quasigroup. Two Latin squares can be combined into a pair of orthogonal Latin squares.