Given a standard chessboard, you can choose any rank or file and repaint each of its squares to the opposite color (white squares turn black, and black squares turn white). By doing this repeatedly, is it possible to produce a board with 63 white squares and one black square?
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No. Repainting a rank or file in this way transforms its k black and (8 – k) white squares into (8 – k) black and k white squares. The number of black squares changes by (8 – k) – k = 8 – 2k. This is an even number, which means that the parity of the number of black squares is preserved. Since we start with an even number (32) of black squares, there is no way to reach the odd outcome 1.
(From Viktor Prasolov’s Problems in Plane and Solid Geometry, 1988.)
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