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Fly on 2 x 1 x 1 brick

Q: A (pointlike) fly is sitting at a corner of a 2 x 1 x 1 brick. What's the furthest point from it, measuring along the surface?
rec.puzzles, 15 Apr 1996

Q: ??? On a centrally-symmetric body, must it be the case that every, or indeed any, longest geodesic's endpoints are antipodes?

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