The 82-queen cover of Q₁₆₃: minimality, and an obstruction at Q₁₆₇

Abstract

The queen's graph Qₙ has the squares of the n × n board as vertices, two squares adjacent when they share a row, a column or a diagonal, and γ(Qₙ) is its domination number. A recent preprint of Kong prints 82 squares of the 163 × 163 board and asserts, on the evidence of a short script, that they dominate it; with the Finozhenok–Weakley bound γ(Qₙ) ≥ ⌈ n/2⌉ this gives γ(Q₁₆₃)=82, an exact value beyond the published list, and the type A structure of the set improves the constant in Östergård–Weakley's amplification theorem to 17/33. We first re-establish every printed property of that certificate by exhaustive verified computation over all 163²=26 569 squares, in place of the script. We then prove two statements the source does not make. The certificate is a minimal dominating set: each of its 82 queens owns a square that no other queen of the set covers, so no 81 of them dominate. And — the main result — it does not extend: with the 82 squares kept in place, no dominating set of Q₁₆₇ of size at most 84 contains them, and none of Q₁₆₅ of size at most 83 does. The first is a nonexistence statement over all 27 889² ordered pairs of added squares, reduced to 499 × 27 889 by an anchor argument whose completeness is proved rather than assumed. Since γ(Q₁₆₇) ≥ 84 by Finozhenok–Weakley, a new exact value at 167 needs a new construction and not an extension of this one. Everything is machine-checked in Lean 4; the two published theorems we rely on — the Finozhenok–Weakley bound and Östergård–Weakley's amplification theorem — are cited rather than reproved, and the Finozhenok–Weakley bound appears as an explicit hypothesis of the statements that use it.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   3a267ae5c06731afa1ad8b59ef3d23094ffce3917250595ef4513c9b3697c3ab