On Turnabout Is Fair Play

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Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board


which we mutilate by removing the square upon which the Baron had placed the tail


Now, at each turn of a pair of coins Sir R----- must place tails upon both one white and one black square, and so if there are more of the one than there are of the other then he cannot possibly succeed in turning them all. The Baron could have ensured that this was the case by placing his tail upon a black square, leaving eleven heads upon the black squares and thirteen of them upon the white squares, and I should therefore have advised Sir R----- to most emphatically decline the wager!
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