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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