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  "body": "\n## Definition\n\nLet $n$ be a positive integer and let $S$ be a [[finite set|0BBTH3YSZA]] with $n$ elements, the symbols.\nA Latin square of order $n$ on $S$ is an $n \\times n$ array $L = (L_{ij})$ with entries in $S$ such that:\n\n1. In each row, each element of $S$ occurs exactly once.\n2. In each column, each element of $S$ occurs exactly once.\n\n## Remarks\n\nExample: for $i, j \\in \\set{0, 1, \\dots, n - 1}$, put $L_{ij} = (i + j) \\bmod n$.\nIn 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.\nThis square is the addition table of the [[integers modulo n|01N3AYAA6N]].\nMore generally, the Cayley table of each finite [[group|G5S1M9]] is a Latin square.\nNon-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.\nA Latin square is the same thing as the multiplication table of a finite quasigroup.\nTwo Latin squares can be combined into a pair of [[orthogonal Latin squares|H6TN6BADVY]].\n",
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