The grasshopper problem

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 1. It then jumps once, a fixed distance d , in a random direction. What shape should the lawn be to maximize 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 < π −1/2 , the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to π ( arcsin ⁡ ( π d / 2 ) ) − 1 . We find transitions to other shapes for d ≳ π − 1 / 2 .