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Winning a strategic number game


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Question

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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  • Solution

The first person with an even number wins.

The last two moves will always be 2->1, 1->0, meaning that the player who changes 3->2 or 4->2 wins. Hence the first player to get 4 wins.

If a player gets an odd number, their only choice is to subtract one, hence forming another even number. Hence a person with an even number can always guarantee themselves another even number on the next move by changing the current even number to an odd number.

So Alice wins if she starts with an even number.

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