New explicit solution to the N-Queens Problem and its relation to the Millennium Problem

TL;DR

This paper presents a novel explicit solution to the N-Queens problem using modular arithmetic, introduces a minimal-width Queen function representation, and explores its implications for the P vs NP Millennium Problem.

Contribution

It provides a new short solution to the N-Queens problem, establishes a minimal width for Queen function representations, and links these findings to the P vs NP Millennium Problem.

Findings

Solutions can be represented with Queen functions of width ≤ 3

The minimal width of Queen functions cannot be further reduced

A conjecture is proposed that, if true, implies polynomial-time solvability of N-Queens completion

Abstract

Using modular arithmetic of the ring $\mathbb{Z}_{n+1}$ we obtain a new short solution to the problem of existence of at least one solution to the $N$-Queens problem on an $N \times N$ chessboard. It was proved, that these solutions can be represented as the Queen function with the width fewer or equal to 3. It is shown, that this estimate could not be reduced. A necessary and sufficient condition of being a composition of solutions a solution is found. Based on the obtained results we formulate a conjecture about the width of the representation of arbitrary solution. If this conjecture is valid, it entails solvability of the $N$-Queens completion in polynomial time. The connection between the $N$-Queens completion and the Millennium $P$ vs $NP$ Problem is found by the group of mathematicians from Scotland in August 2017.