Where have all the grasshoppers gone?
J\'anos Pach, G\'abor Tardos

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 be an -element point set in the plane. Consider (pointlike) grasshoppers sitting at different points of . 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 ? 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 is the vertex set of a regular -gon and . Some generalizations are also considered.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Mathematics and Applications · graph theory and CDMA systems
