No Attraction

http://commons.wikimedia.org/wiki/File:Kepler%27s_law_2_ru.svg

Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.

If gravity were turned off, would this still be true?

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The Suicidal King

Imagine a 1000 x 1000 chessboard on which a white king and 499 black rooks are placed at random such that no rook threatens the king. And suppose the king goes bonkers and wants to kill himself. Can he reach a threatened square in a finite number of moves if Black is trying actively to avoid this?

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