Example: coins are arranged in a square. Initially, all coins are heads up. Each time, five adjacent coins (either horizontally, vertically, or diagonally) are flipped.
After a finite number of such flips, is it possible for all the coins to be tails up?
Problem 1087
Official solution
It is impossible. As follows, each coin is marked, then:
(1) The five coins flipped each time are , , , , .
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\end{tabular}
(2) At the same time, for each mark , , , , there are 58 coins, and for there are 57 coins.
(3) If all the coins are finally face down, then each coin must have been flipped an odd number of times.
From (2) and (3), it is known that all the coins marked have been flipped an even number of times, and all the coins marked have been flipped an odd number of times. Therefore, the number of times the coins marked are flipped is different from the number of times the coins marked are flipped, which contradicts (1). Therefore, it is impossible.