Hidden Structure in the Solutions Set of the N Queens Problem

TL;DR

This paper explores the hidden geometric and algebraic structures within the solution set of the N Queens problem, revealing connections to Boolean quadratic forms, matrices, and fractal operators, indicating complex underlying patterns.

Contribution

It introduces a novel framework linking N Queens solutions to Boolean quadratic forms and fractal operators, uncovering new structural insights.

Findings

Solutions are encoded in a special matrix.

A connection to Boolean fractal operators is established.

Evidence of underlying geometric structure is found.

Abstract

Some preliminary results are reported on the equivalence of any n-queens problem with the roots of a Boolean valued quadratic form via a generic dimensional reduction scheme. It is then proven that the solutions set is encoded in the entries of a special matrix. Further examination reveals a direct association with pointwise Boolean fractal operators applied on certain integer sequences associated with this matrix suggesting the presence of an underlying special geometry of the solutions set.