{
  "schema_version": 1,
  "problem_number": "AMR-123-0003",
  "title": "A Partial Answer to Propp's Problem 7 on Benzel Tilings: An Infinite Family of Type-103 Tilings and the Open Cases with |a−b| ≤ 7 or max(a,b) ≤ 60",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Propp asked whether every (a,b)-benzel with nonnegative Conway–Lagarias invariant has a tiling by right-pointing stones and bones of all three orientations, a tiling of type 103 (Problem 7 of his list of twenty problems). By published theorems of Defant, Li, Propp and Young, of Defant, Foster, Li, Propp and Young, and of Kim and Propp, the question is open only for the benzels (d+3J, 2d+3J) with d ≥ 3 and 1 ≤ J ≤ (d−2)(d+1)/2 and for their mirror images. We give a partial answer; Problem 7 remains open in general. We prove that for every D ≥ 2 the (4D−3, 5D−3)-benzel has a type-103 tiling, formed by a central triangle of (D−1)(D−2)/2 right stones and three congruent sectors, each tiled by parallel bones; for D ≥ 3 these benzels lie in the open region. We also verify the question, by computer, for every open case with |a−b| ≤ 7 and for every open case with max(a,b) ≤ 60: for each of these benzels a tiling is stored and checked by a short self-contained program. This is a finite verification and says nothing about larger cases. The row (n, 2n−3) and the window max(a,b) ≤ 14 are covered by unrefereed preprints of 2026 (by A. C. Li and by The Clankers), which we credit and do not use. We also record computational evidence that the simplest ring inductions, which add a ring around a smaller benzel, do not reach the cases near the line where the invariant vanishes. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "benzels",
    "trihex tilings",
    "stones and bones",
    "trimer covers",
    "triangular grid",
    "Conway-Lagarias invariant",
    "type-103 tilings",
    "Propp's problems",
    "partial answer",
    "computer-assisted verification",
    "UnsolvedMath",
    "AMR-123-0003",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-123-0003/",
  "pdf_url": "https://eulersolve.org/papers/amr-123-0003/paper.pdf?v=db54a5ddc57a",
  "doi": "10.5281/zenodo.23483633",
  "zenodo_record_url": "https://zenodo.org/records/23483633",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Propp's Problem 7: Problem 7 stays open in general; the row (n, 2n-3) and the window max(a,b) <= 14 are covered by unrefereed 2026 preprints, which are credited and not used.",
  "files": {
    "paper.pdf": {
      "sha256": "db54a5ddc57a3d5c1b58043763567a5754cb8031f19db36ca3788f118ca9a537"
    },
    "source.zip": {
      "sha256": "ece47e611727ff5ce177e9ecd63cc6030432d28f81fe0c1ce9131faef4daf5e2"
    },
    "verification_report.md": {
      "sha256": "a3632a07ab45285ca77c87acdbdb9b3ce351500082011c7de6bfea276116731d"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}