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Article
The Grasshopper Problem
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
(2017)
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 < 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 .
Keywords
- geometric combinatorics,
- spin models,
- Bell inequalities,
- statistical physics
Disciplines
Publication Date
November, 2017
DOI
10.1098/rspa.2017.0494
Citation Information
Olga Goulko and Adrian Kent. "The Grasshopper Problem" Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Vol. 473 (2017) p. 20170494 Available at: http://works.bepress.com/olga-goulko/3/