  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 $d \gtrsim
\pi^{-1/2}$.
