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July Puzzle Periodical - First or Second?

By Sean A., NSA Applied Mathematician

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On a rainy summer day, brothers Dylan and Austin spend the day playing games and competing for prizes as their grandfather watches nearby. After winning two chess matches, three straight hands of poker and five rounds of ping-pong, Austin decides to challenge his brother, Dylan, to a final winner-take-all competition. Dylan clears the kitchen table and Austin grabs an old coffee can of quarters that their dad keeps on the counter.

The game seems simple as explained by Austin. The brothers take turns placing a quarter flatly on the top of the square kitchen table. Whoever is the first one to not find a space on his turn loses. The loser has to give his brother tonight’s dessert. Right before the game begins, Austin arrogantly asks Dylan, “Do you want to go first or second?”

Dylan turns to his grandfather for advice. The grandfather knows that Dylan is tired of losing every game to his brother. What does he whisper to Dylan?

Click to see the answer!

Solution

Dylan should go first. By doing this, Dylan can guarantee a win by playing to a deliberate strategy. On his first turn, he can place a quarter right on the center of the table. Because the table is symmetric, whenever Austin places a quarter on the table, Dylan simply "mirrors" his brother’s placement around the center quarter when it is his turn. For example, if Austin places a quarter near a corner of the table, Dylan can place one on the opposite corner. This strategy ensures that even when Austin finds an open space, so can Dylan. As a result, Dylan gains victory, since Austin will run out of free space first!