A puzzle by Y. Ionin, from the September/October 1990 issue of Quantum:
Three frogs occupy three vertices of a square. When one frog jumps over another, it lands beyond it at the same distance that had originally separated them. Can any frog reach the fourth vertex?
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No. Impose coordinates so that the frogs sit at (0,0), (1,0), and (0,1). Now when a frog at (x, y) jumps over a frog at (a, b), it lands at (2a – x, 2b – y). Thus the parities of each frog’s coordinates never change. At the start each frog had at least one even coordinate, so none of them can ever reach a point with two odd coordinates, such as (1,1).
See the issue, Solution M14 on page 59, for a visual proof.
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