Coins on a table Death, having given up on
chess, decided to play a new game to taunt mortals with. He has a
rectangular table and you take turns to place coins on the table. The last one
to be able to lay down a coin wins the game. I don't need to tell you what's
in store if you lose!
The Puzzle: The extraordinary claim is that one of the players (i.e. either the player to start or the second player) has a winning strategy. Determine who has the winning strategy and what it is.
Further thoughts: It isn't necessary that the game is played on a rectangular table. After finding the solution, you should be able to determine the class of tables for which the winning strategy always works.
Hints and solutions below:
Hints will appear here!
To find out who has the winning strategy, think about what would happen if the coins were huge, almost as big as the table. Note when we ask for the winning strategy, we want you to give a precise description of what the player should do in response to his (or her) opponent's move.
Exploit the symmetry of the situation.
The first player has the winning strategy (this is clear if the coins are as big as the table!) You start by placing your coin in the middle of the table, establishing your supremacy in the game of symmetry. No matter what move your opponent makes next, you can always place a coin at the diametrically opposite point. The game continues in this way, with you always choosing to play the opposite position to your opponent.
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