{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nWythoff discovered that the cold positions follow a regular pattern determined by the [[golden ratio|0SC8T7548D]]. Specifically, if *k* is any natural number and\n\n$$n_k = \\lfloor k \\phi \\rfloor = \\lfloor m_k \\phi \\rfloor -m_k \\,$$\n\n$$m_k = \\lfloor k \\phi^2 \\rfloor = \\lceil n_k \\phi \\rceil = n_k + k \\,$$\n\nwhere φ is the golden ratio and we are using the floor function, then (*n*\u003csub\u003e*k*\u003c/sub\u003e, *m*\u003csub\u003e*k*\u003c/sub\u003e) is the *k*\u003csup\u003eth\u003c/sup\u003e cold position. These two sequences of numbers are recorded in the Online Encyclopedia of Integer Sequences as and , respectively.\n\nThe two sequences *n\u003csub\u003ek\u003c/sub\u003e* and *m\u003csub\u003ek\u003c/sub\u003e* are the Beatty sequences associated with the equation\n\n$$\\frac{1}{\\phi} + \\frac{1}{\\phi^2} = 1 \\,.$$\n\nAs is true in general for pairs of Beatty sequences, these two sequences are [[complementary|0RKF1ZT5JJ]]: each positive integer appears exactly once in either sequence.\n",
  "commit": "135c6437fad4260afa9c407aa7649bf9dea295b7",
  "front": {
    "inventory": [
      "P1XEMQS1GM"
    ],
    "kind": "statement",
    "name": "Formula for cold positions",
    "role": "proposition",
    "sources": [
      {
        "locator": "Wythoff's game, section Formula for cold positions",
        "mention": "wikipedia://en/10222175#BF5Z",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:10222175@r1338739976"
      }
    ],
    "tag": "8HKFHQPX13"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/8HKFHQPX13/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8HKFHQPX13.md",
  "msc": null,
  "name": "Formula for cold positions",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8HK/8HKFHQPX13.md",
  "status": null,
  "tag": "8HKFHQPX13",
  "uri": "m2://8HKFHQPX13",
  "url": "https://math2.tamnd.com/wiki/formula-for-cold-positions/",
  "writing": {
    "tag": "8HKFHQPX13",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}