# The grasshopper problem

## TL;DR

This paper investigates the optimal shape of a lawn to maximize a grasshopper's probability of remaining on it after a jump, revealing that a disc is not optimal and identifying cogwheel shapes as solutions for certain jump distances.

## Contribution

It introduces a geometric combinatorics problem inspired by Bell inequalities and models it using a spin system to identify optimal lawn shapes.

## Key findings

- Disc is not optimal for any jump distance d>0.
- Cogwheel-shaped lawns are optimal for small d, with the number of cogs depending on d.
- Transitions to other shapes occur as d increases.

## Abstract

We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area one. It then jumps once, a fixed distance $d$, in a random direction. What shape should the lawn be to maximise the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc shaped lawn is not optimal for any $d>0$. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that for $ d < \pi^{-1/2}$ the optimal lawn resembles a cogwheel with $n$ cogs, where the integer $n$ is close to $ \pi ( \arcsin ( \sqrt{\pi} d /2 ) )^{-1}$. We find transitions to other shapes for $d \gtrsim \pi^{-1/2}$.