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 Puzzle Archive Puzzle I You play this game with your friend. Starting with an empty rectangular table, and an unlimited supply of quarters, each person takes a turn by putting a quarter anywhere on the table. The rules say only that you must place your quarter so that it doesn't touch any quarter already on the table. You and your opponent take turns placing quarters in succession, until the table is nearly full of quarters. The first player who is unable to add a quarter without touching a quarter already placed loses. You move first. What strategy would you use to play this game? Solution: The first move is often an advantage. In ticktacktoe, you want to occupy the center square on your first move. There's an infinity of possible positions in which to place the first quarter. This is evidently a game of many moves (it takes a lot of quarters to cover a table so densely that you can't add another coin without touching one already placed). It might be that the first player can secure an advantage that persists throughout the game. And maybe not. No matter what you do on your first move, your opponent seems to be able to effectively duplicate it on her first move. All she has to do is place her quarter at a position 180 degrees rotated from your quarter. If you take the northeast corner, she takes the southwest corner, and so on. There is one exception only — a move your opponent can't duplicate. That is to put your first quarter in the exact center of the table. There is only one center of the table. Once covered by a quarter, the spot cannot be taken by anyone else. It is simply the only unique first move, the only one that exploits your status as first player to do something your opponent will not be able to copy. No matter what you do, the other player has freedom to put her quarters almost anywhere on the table throughout the early phase of the game. Now you have to just "mirror" your opponent's previous move. Draw a line from her last quarter, through the center of the table, and beyond. Place your next quarter on the extension of that line, exactly as far from the center as the opponent's quarter (but on the opposite side of the center).You will always be able to do this, for you are only duplicating your opponent's last action (given the symmetry of the table). Eventually, your opponent will be the one who's unable to place an additional quarter without touching one already placed. |
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