A grasshopper lands at a random point on a planar lawn of area one. It then makes one jump of 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? This easily stated yet hard to solve mathematical problem has intriguing connections to quantum information and statistical physics. A generalized version on the sphere can provide insight into a new class of Bell inequalities. A discrete version can be modeled by a spin system, representing a new class of statistical models with fixed-range interactions, where the range d can be large. I will show that, perhaps surprisingly, there is no d > 0 for which a disc shaped lawn is optimal. If the jump distance is smaller than the radius of the unit disc, the optimal lawn resembles a cogwheel, with transitions to more complex, disconnected shapes at larger d. Using parallel tempering Monte Carlo for the discrete spin model, several classes of optimal lawn shapes with different symmetry properties can be identified.