{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nThirty-six officers come from six regiments, with one officer of each of six ranks in each regiment.\nArrange them in a $6 \\times 6$ square with two conditions.\nEach row and each column contains exactly one officer of each regiment and exactly one officer of each rank.\n\n## Remarks\n\nWriting the regiment and the rank of the officer in each cell gives two Latin squares of order $6$.\nThe arrangement asked for exists if and only if these two squares are [[orthogonal|KHD2QMNR7R]].\nSo the problem asks for a pair of orthogonal Latin squares of order $6$.\nEuler posed the problem in 1782 and believed it had no solution.\nGaston Tarry proved in 1901, by a case analysis, that there is no solution.\nEuler also conjectured that no orthogonal pair exists for any order $n \\equiv 2 \\pmod 4$.\nBose, Shrikhande and Parker disproved this around 1959 to 1960: an orthogonal pair of order $n$ exists for every $n$ except $2$ and $6$.\n\n## Explanations\n\n### L1 human\n\nThe puzzle is a two-colour Sudoku without boxes: every row and column must show each regiment once and each rank once.\nFor $25$ officers in a $5 \\times 5$ square the analogous puzzle has a solution, but for $36$ officers it has none.\nIt is surprising that $6$ fails while $10$, $14$ and every larger order work.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "H6TN6BADVY"
    ],
    "history": {
      "first_stated": "1782",
      "posed_by": "Leonhard Euler"
    },
    "kind": "problem",
    "msc": [
      "05B15"
    ],
    "name": "Euler's thirty-six officers problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares"
      }
    ],
    "tag": "WXJAVQKFR6",
    "uses": [
      "KHD2QMNR7R"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/WXJAVQKFR6/",
  "links": [
    {
      "field": "about",
      "target": "H6TN6BADVY"
    },
    {
      "field": "uses",
      "target": "KHD2QMNR7R"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WXJAVQKFR6.md",
  "msc": [
    "05B15"
  ],
  "name": "Euler's thirty-six officers problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WXJ/WXJAVQKFR6.md",
  "status": null,
  "tag": "WXJAVQKFR6",
  "uri": "m2://WXJAVQKFR6",
  "url": "https://math2.tamnd.com/wiki/eulers-thirty-six-officers-problem/",
  "writing": {
    "tag": "WXJAVQKFR6",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}