{
  "abstract": "Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order $+,\\times,-,÷$. We ask for the fewest moves needed to make every integer from $1$ to $N$. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least $N-1$ moves are needed. We show that $N-1$ moves are also enough for every $N\\ge 9$. Conventional induction cannot work, because a division that comes right after a completed interval $\\{1,\\dots,P\\}$ produces numbers already made. Instead we extend a completed interval $\\{1,\\dots,P\\}$ to $\\{1,\\dots,3P\\}$ all at once, which counting shows is the smallest multiplicative extension $P\\to kP$ that can work, and then adjust the last few moves to reach every other $N$. For $9\\le N\\le 33$ we give explicit sequences, found by computer search.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10095",
  "author_total": 2,
  "authors": [
    {
      "name": "Sean Lesmana"
    },
    {
      "name": "Theodore Tjugiarto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/28mbttwdsg51t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10095",
  "fulltext_url": null,
  "id": "28mbttwdsg51t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10095",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220955295",
  "pdf_url": "https://arxiv.org/pdf/2610.10095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/28mbttwdsg51t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10095"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Making Every Number from 1 to N Under a Fixed Cycle of $+$, $\\times$, $-$, $÷$",
  "title_text": "Making Every Number from 1 to N Under a Fixed Cycle of +, ×, −, ÷",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/28mbttwdsg51t",
  "venue": "arXiv",
  "versions": []
}