Where have all the grasshoppers gone?

TL;DR

This paper investigates whether grasshoppers on a plane can rearrange themselves through specific jumps to form a larger set, providing a linear algebraic solution for certain configurations like regular polygons.

Contribution

It introduces a linear algebraic method to determine the possibility of grasshopper rearrangements and solves a problem for regular polygons with specific sizes.

Findings

Rearrangement is possible for regular polygons with N ≠ 3, 4, 6.

Provides a linear algebraic framework for the problem.

Extends the analysis to some general configurations.

Abstract

Let $P$ be an $N$-element point set in the plane. Consider $N$ (pointlike) grasshoppers sitting at different points of $P$. In a "legal" move, any one of them can jump over another, and land on its other side at exactly the same distance. After a finite number of legal moves, can the grasshoppers end up at a point set, similar to, but larger than $P$? We present a linear algebraic approach to answer this question. In particular, we solve a problem of Brunck by showing that the answer is yes if $P$ is the vertex set of a regular $N$-gon and $N\neq 3, 4, 6$. Some generalizations are also considered.