{
  "abstract": "Smith's remoteness function records how long an impartial game lasts when the winner wins as quickly as possible and the loser delays defeat. For the generalized Wythoff game $\\mathrm{WYT}(a,b)$, in which two positive integer parameters $a$ and $b$ determine which simultaneous removals from the two piles are allowed, we give a formula for the remoteness of every position, building on the known values at losing positions. The formula depends on two indices: a local index determined by the coordinates of the losing positions in a window of length $b$ ending at the smaller pile, and the integer part of the pile difference divided by $a$. When $b=1$, complementary Beatty sequences make the local index explicit, which gives an explicit formula for $\\mathrm{WYT}(a,1)$, including classical Wythoff's game. We also show that if each attempted move in a finite impartial game independently fails with a fixed probability and passes the turn, the probability that the player to move wins under optimal play is an explicit exponential function of remoteness. For every success probability strictly between $0$ and $1$, the optimal attempts are exactly the moves that reduce remoteness by one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30427",
  "author_total": 2,
  "authors": [
    {
      "name": "Raúl Sánchez Galán"
    },
    {
      "name": "Yijie Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0gtv1xqpm9qxt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30427",
  "fulltext_url": null,
  "id": "0gtv1xqpm9qxt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30427",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223321984",
  "pdf_url": "https://arxiv.org/pdf/2610.30427",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0gtv1xqpm9qxt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30427",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30427"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A remoteness formula for generalized Wythoff games",
  "title_text": "A remoteness formula for generalized Wythoff games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/0gtv1xqpm9qxt",
  "venue": "arXiv",
  "versions": []
}