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 maximize the chance that the grasshopper lands on the lawn again?
Surprisingly, the lawn should never be shaped like a disk! Here's what it should look like for various choices of 𝑑. For larger values of 𝑑 it gets even weirder.
