# The grasshopper problem

• Published in 2017
In the collections
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 $d \gtrsim \pi^{-1/2}$.

## Other information

key
Thegrasshopperproblem
type
article
2018-01-24
date_published
2017-07-24

### BibTeX entry

@article{Thegrasshopperproblem,
key = {Thegrasshopperproblem},
type = {article},
title = {The grasshopper problem},
author = {Olga Goulko and Adrian Kent},
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{\}}{\$}.},
comment = {},
}