{
  "abstract": "We study a one-player token game on an $N\\times N$ board where tokens slide along diagonals or duplicate onto neighbouring ones to form a combinatorial rectangle $R\\times S$. A conserved integer weight $W'$ and a strict monovariant guarantee $O(N^2)$-length solutions, placing the game in $\\mathsf{NP}$. We prove that reaching a final position factors this $2N$-bit $W'$ into two $N$-bit factors $V,M < 2^N$ that encode the rectangle's rows and columns. Consequently, solving the game for a balanced-semiprime target is equivalent to integer factoring. However, if the target rectangle is known, the solution reduces to two polynomial-time steps: a forced downward chip-flow and a $0/1$-polynomial factorisation leveraging Cohn's theorem. The game's entire difficulty is thus isolated to the initial number-theoretic split. Supplying the popcounts of the factors as a promise preserves this asymptotic hardness but bounds the target search space. We exploit this constrained space using a learned policy/value network and an AlphaZero-style Monte-Carlo tree search, empirically probing the limits of neural look-ahead on a factoring-equivalent environment.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22968",
  "author_total": 1,
  "authors": [
    {
      "name": "Marcel Crasmaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fzhbaq3mwdjm6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22968",
  "fulltext_url": null,
  "id": "fzhbaq3mwdjm6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22968",
  "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",
      "cs.CR"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Artificial Intelligence in Games"
    ]
  ],
  "openalex": "W7213996572",
  "pdf_url": "https://arxiv.org/pdf/2609.22968",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fzhbaq3mwdjm6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22968",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22968"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Searching for Primes: A Neural AlphaZero Approach to a Factoring Game",
  "title_text": "Searching for Primes: A Neural AlphaZero Approach to a Factoring Game",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fzhbaq3mwdjm6",
  "venue": "arXiv",
  "versions": []
}