Alice and Bob play a number game. Starting with a positive integer n, they take turns changing the number with Alice as the first player. Each player in turn may change the number n to a new positive integer k or zero. either by k -1 or k/2. The person who changes 1 to 0 wins. For instance, when n = 3, the players have no choice, k proceeds from 3 to 2 to 1 to 0, and Alice wins. When n = 4, Alice wins if and only if her first move is to change 4 to 2. For which initial n does Alice have a winning strategy?

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## BMAD 64

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