Fleas on a Chessboard

In every square of a 5×5 board there is a flea. At some point, all the fleas jump to an adjacent square (two squares are adjacent if they share an edge). Is it possible that after they settle in the new squares, the configuration is exactly as before: one flea per square?

Solution


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Copyright © 1996-2017 Alexander Bogomolny

In every square of a 5×5 board there is a flea. At some point, all the fleas jump to an adjacent square (two squares are adjacent if they share an edge). Is it possible that after they settle in the new squares, the configuration is exactly as before: one flea per square?

Solution

Assume that corner squares are black. In a 5×5 board, there 13 black and 12 white square. In a jump, the fleas change the color of their square. Therefore, 13 fleas that start on the black squares are bound to land on at most 12 white squares, meaning that at least in one square there will be at least two fleas.

From the "white" fleas perspective, these 12 fleas land on 13 black squares, meaning that at least one of the black squares is bound to be empty.


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