Minimal number of points on a grid forming line segments of equal length

TL;DR

This paper investigates the minimal number of grid points needed to form a specified number of equal-length line segments, relating it to the n-queens problem and providing bounds based on number theory.

Contribution

It introduces a novel connection between grid point configurations for equal-length segments and the n-queens problem on toroidal boards, establishing new bounds.

Findings

Upper bound of kn/3 for the number of points

Asymptotic approach to kn/4 when conditions are met

Relation to the n-queens problem on toroidal chessboards

Abstract

We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that this number is upper bounded by $kn/3$ and approaches $kn/4$ as $n\rightarrow\infty$ when $k+1$ is coprime with $6$ or when $k$ is large.