MMM25.pdf
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… following three games, players take turns. That is, player 1 starts, then player 2, then player 1, and so on. For each of the following three games, determine which player has a winning strategy: player 1, player 2, or neither. If there is a player with a winning strategy, then describe the winning strategy. Game 1 Player 1 starts by saying the number 1, then player 2 says the number 2, and the one who’s next must say an integer strictly between the previous number and twice of it (not including … 1 starts by saying 1, then player 2 says 2. Player 1’s options are now all integers strictly between 2 and 4, but there’s only one such option: 3. So player 1 says 3. Player 2’s options are now between 3 and 6, which are 4 and 5. The game finishes when someone says 100 or larger; that player wins. Game 2 Consider a row of n ≥ 2 coins of values v1, . . . , vn, where n is even. Assume that v1 + v2 + . . . + vn is odd. For example: 7 9 4 3 3 5 In each turn, a player selects either the first or last … 5. For the weekend, Alice bought three apple pies, to share with her visitors Bob and Carol. (a) On Friday evening Bob is there, but Carol is stuck in the traffic. Hav- ing waited for two hours Carol still hasn’t arrived. So Alice and Bob decide they go ahead with the first apple pie anyway. First Alice takes half of the apple pie. Then Bob takes half of the half that remains. Then Alice takes half of what is still left. Then Bob takes half of what is left. And so they continue, on and on, …