{
  "schema_version": 1,
  "problem_number": "OWR-14299905-036",
  "title": "The Word-Search Constants C_2(ABBB) = 8/5 and C_3(ABB) = 6: On Two Questions of Schildkraut and of Halberstam and Schildkraut",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a word w and a toroidal grid of letters in dimension d, an appearance of w is a cell together with one of the 3^d − 1 directions in {−1,0,1}^d ∖ {0} along which the letters spell w, and C_d(w) is the supremum, over all grids, of the number of appearances per cell. We prove that C_2(ABBB) = 8/5. This answers Question 9.3 of Halberstam and Schildkraut (Words with repeated letters in a grid, 2025) and the last part of Question 11 of Schildkraut in the problem session of the Oberwolfach workshop Combinatorics (Oberwolfach Rep. 23 (2026)). We also prove that C_3(ABB) = 6, which is the case d = 3 of the first part of that Question 11 and of their Question 9.2. Both values also hold for n × … × n arrays without wraparound, as in the report. The lower bounds are known constructions. The upper bounds are computer-assisted: averaging lemmas, proved in the text, reduce them to 65,536 and 2^27 integer inequalities, one for each pattern of a 4 × 4 or a 3 × 3 × 3 window; these are given by exact certificates and were checked in integer arithmetic by independent programs. In both cases we classify the extremal grids. For ABBB they are the two five-queens lattices, on tori with both sides divisible by 5; for ABB in dimension 3 they are the grids in which the letters A form a family of parallel planes modulo 3. Every window that is not of this kind costs at least 3/70, respectively 47137/294912. So for these two pairs (d, w), and only for these, we show that the supremum is attained by a periodic grid; Question 10 of the report (Question 2.5 of Halberstam and Schildkraut) asks this for all pairs. For ABBB we also determine the possible pairs (density of A, concentration): their closure is the quadrilateral with vertices (0,0), (1/5, 8/5), (3/8, 3/2), (1,0). It follows that Question 9.6 of Halberstam and Schildkraut, a stability statement with constant 1 for the letter distribution of near-extremal grids, has a negative answer for the pair (ABBB, 2) as stated; it holds with the optimal constant 14/5 for grids on two letters. Finally 18 ≤ C_4(ABB) ≤ 62/3, 54 ≤ C_5(ABB) ≤ 64 and 24/5 ≤ C_3(ABBB) ≤ 11/2. Question 10 in general and the values of C_d(ABB) for d ≥ 4 remain open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "keywords": [
    "word search",
    "words in grids",
    "toroidal grids",
    "concentration of a word",
    "stackable words",
    "five-queens lattice",
    "extremal grids",
    "stability",
    "linear programming certificates",
    "computer-assisted proof",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299905-036",
    "math.CO",
    "cs.DM",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299905-036/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299905-036/paper.pdf?v=8d1ef49962ad",
  "doi": "10.5281/zenodo.23282536",
  "zenodo_record_url": "https://zenodo.org/records/23282536",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Questions 10 and 11 of C. Schildkraut (Oberwolfach Reports 23 (2026), Report 1/2026, problem session of the workshop Combinatorics; Z. Halberstam and C. Schildkraut, arXiv:2511.19678): the note determines C_2(ABBB) = 8/5 and C_3(ABB) = 6 and classifies the extremal grids in these two cases. The upper bounds are computer-assisted: the reductions are proved in the text, and the finite inequalities (65,536 and 2^27 patterns) are given by exact certificates which were verified in integer arithmetic by independent programs. Question 10 in general and the values of C_d(ABB) for d ≥ 4 remain open.",
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  "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."
}