# Question :

Consider a two-player game played on a circular table of unspecified diameter.

Each player has an infinite supply of coins, and take turns placing a coin on the table such that it is completely on the table and does not overlap with any other coins already played.

A player wins if he makes the last legal move.

Which player (if any) has a strategy that will guarantee a win? Assume that the diameter of the table is greater than the diameter of the coin.

# Solution :

Player 1 has a strategy to win. His first move is to put a quarter in the exact center of the table.

Player 2 then places a quarter anywhere on the table (but not in the center, of course). Now convince yourself that Player 1 always has a legal move: by playing a quarter diametrically opposite from the quarter Player 2 just put down (i.e., Player 1’s first quarter is the midpoint of the line between Player 2’s last quarter and Player 1’s last quarter). We continue until we fill up the table, but since Player 1 always has a legal move, it is only Player 2 that will be faced with no legal move. Then Player 1 wins.

Note that this strategy works even in the degenerate case when the table is the size of a quarter.

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