## prsa.suppl.material.description from The grasshopper problem

media

posted on 07.11.2017 by Olga Goulko, Adrian Kent#### media

Media is any form of research output that is recorded and played. This is most commonly video, but can be audio or 3D representations.

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 (√*πd*/2))^{−1}. We find transitions to other shapes for*d*≳*π*^{−1/2}.