Maximum arrangements of nonattacking kings on the $2n\times 2n$ chessboard

TL;DR

This paper develops a recursive method and formula for counting maximum arrangements of nonattacking kings on a 2n by 2n chessboard, using matrix constructions based on independent arrangements of smaller blocks.

Contribution

It introduces a novel recursive approach and explicit formula for counting maximum nonattacking king arrangements on large chessboards.

Findings

Derived a recursive construction for arrangements

Provided explicit formulas and algorithms

Achieved bounds for maximum arrangements

Abstract

To count the number of maximum independent arrangements of $n^2$ kings on a $2n\times 2n$ chessboard, we build a $2^n \times (n+1)$ matrix whose entries are independent arrangements of $n$ kings on $2\times 2n$ rectangles. Utilizing upper and lower bound functions dependent of the entries of the matrix, we recursively construct independent solutions, and provide a straight-forward formula and algorithm.