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Lawnmower

Lawnmower

Q: ??? An automatic lawnmower works by moving in a straight line until it hits an edge of the lawn, turning through a fixed angle, and starting again (in fact it should turn through the angle repeatedly until it's heading back onto the lawn). Say it succeeds if it eventually passes within epsilon of every point of the lawn. Does there exist a lawn (and a starting point) for which it fails? For rational angles the answer is yes - pick a suitable polygon - but what about irrational angles? What if we insist the lawn is simply connected? Convex?
[TB, 12 Jul 1996]

A: If the angle is less than 90 degrees then it can never reach the centre of a circular lawn [Barbara Davies].

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