John Carlos Baez on Nostr: A grasshopper lands at a random point on a planar lawn of area one. It then jumps ...

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 remains on the lawn when it lands?

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.

• Olga Goulko and Adrian Kent, The grasshopper problem, https://arxiv.org/abs/1705.07621.

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