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Alice and Bob play the following game. A stack of n tokens lies before them. The players take turns with Alice going first. On each
Alice and Bob play the following game. A stack of n tokens lies before them. The players take turns with Alice going first. On each turn, the player either removes token or tokens from the stack. Whoever removes the last token wins. Find the number of positive integers n less than or equal to for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alices play. Work out the final answer.
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