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
.