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On Turnabout Is Fair Play

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...

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On Turning Sixteen

In his last wager, the Baron challenged Sir R----- to put a four by four square of tiles, numbered from one to sixteen, into sequential order from top-left to bottom-right by rotating two by two squares of adjacent tiles. That Sir R----- should have had a prize of three coins if he had succeeded and the Baron should have had one of two coins if he had not suggests that the key to reckoning whether Sir R----- should have taken this wager is to figure whether or not every arrangement of the tiles might be arrived at through such manipulations.

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On Onwards And Upwards

In the Baron's latest game, Sir R----- was to place a queen upon the first rank or file of a chessboard and the Baron was to then move it horizontally, vertically or diagonally in its regular manner to a square of higher rank and/or file. Sir R----- was then to do the same in his turn and, alternating thereafter, he who moved the queen to the top-rightmost square was to be declared the winner.

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On Endarkenment

The first thing to note about the Baron's game is that since four tiles must be turned over at each turn it is impossible to turn out an odd number of lamps. If one of the four was lit at the start of the turn, then three would be at its end yielding a net change of two. Considering the change made when from zero to four tiles were lit at the outset makes it clear that a game with an odd parity of lit lamps cannot be won.
If presented with such a board Sir R----- should most certainly have declined the Baron's wager.

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