{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "KHD2QMNR7R"
    },
    {
      "field": "about",
      "target": "WXJAVQKFR6"
    }
  ],
  "body": "\n## Summary\n\nTwo [[Latin squares|CDVRMVED2V]] of the same order are orthogonal if, when one is laid over the other, every ordered pair of symbols appears exactly once.\nA family of Latin squares is mutually orthogonal if every two of them are orthogonal.\nFor $n \\ge 2$, there are at most $n - 1$ mutually orthogonal Latin squares of order $n$.\nA family of $n - 1$ of them exists exactly when a [[finite projective plane|2HWB7P1DZN]] of order $n$ exists.\nThe node [[mutually orthogonal Latin squares|KHD2QMNR7R]] gives the precise definition.\n\n## Explanations\n\n### L1 human\n\nImagine a $3 \\times 3$ field planted with three crops and treated with three fertilisers.\nEach crop should appear once per row and column, and so should each fertiliser.\nIf in addition every crop meets every fertiliser exactly once, the two Latin squares are orthogonal.\nSuch a pair exists for every order except $2$ and $6$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "05B15"
    ],
    "name": "Mutually orthogonal Latin squares",
    "same_as": {
      "wikidata": "Q615342",
      "wikipedia": "Mutually orthogonal Latin squares"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares"
      }
    ],
    "tag": "H6TN6BADVY"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/H6TN6BADVY/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/H6TN6BADVY.md",
  "msc": [
    "05B15"
  ],
  "name": "Mutually orthogonal Latin squares",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H6T/H6TN6BADVY.md",
  "status": null,
  "tag": "H6TN6BADVY",
  "uri": "m2://H6TN6BADVY",
  "url": "https://math2.tamnd.com/wiki/mutually-orthogonal-latin-squares/",
  "writing": {
    "tag": "H6TN6BADVY",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}