An Explicit 82-Queen Covering of the 163 x 163 Board and Its Asymptotic Implication

Abstract

The queen's graph $Q_n$ has the squares of the $n\times n$ chessboard as vertices, with adjacency defined by a common row, column, or diagonal. We give an explicit set of $82$ queens on $Q_{163}$. In centered coordinates, all queen coordinates are odd, every odd row and odd column is occupied exactly once, and the occupied rows, columns, and diagonals satisfy the conditions for a type A $1$-cover in the terminology of Ostergard and Weakley. A direct independent verification checks every one of the $163^2=26{,}569$ board squares and finds none uncovered. Hence $\gamma(Q_{163})\leq82$. The Finozhenok-Weakley lower bound $\gamma(Q_n)\geq\lceil n/2\rceil$, valid here, gives the matching inequality and therefore $\gamma(Q_{163})=82$. For this cover, the parameters defined by Ostergard and Weakley are $e=16$, $f=15$, and $u=24$; the complete difference- and sum-diagonal multisets are displayed in the paper. It consequently also supplies an explicit finite input to their amplification theorem for type A covers, giving $\gamma(Q_N)\leq(17/33)N+O(1)$. This last coefficient improves both the earlier type A coefficient $69/133$ and the subsequent general coefficient $101/195$ of Burger and Mynhardt. The coordinates and a complete standard-library Python verifier are included.