2024 AIME I Problems/Problem 3

Problem

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 removes either 1 token or 4 tokens from the stack. Whoever removes the last token wins. Find the number of positive integers n less than or equal to 2024 for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.

Solution 1

Let's first try some experimentation. Alice obviously wins if there is one coin. She will just take it and win. If there are 2, then Alice will take one and then Bob will take one. If there are 3, Alice will take 1, Bob will take one, and Alice will take the final one. If there are 4, Alice will just remove all 4 at once. If there are 5, no matter what Alice does, Bob can take the final coins in one try. Notice that Alice wins if there are 1, 3, or 4 coins left. Bob wins if there are 2 or 5 coins left. After some thought, you may realize that there is a strategy for Bob. If he can just ensure that there are 5 coins remaining, he wins. Any multiple of 5 will also work. The reason for this is the following: Let's say there are a multiple of 5 coins remaining in the stack. If Alice takes 1, Bob will take 4, and there will still be a multiple of 5. If Alice takes 4, Bob will take 1, and there will still be a multiple of 5. This process will continue until you get 0 coins left.

After some more experimentation, you'll realize that any number that is congruent to 2 mod 5 will also work. This is because Bob can do the same strategy, and when there are 2 coins left, Alice is forced to take 1 and Bob takes the final one.

So we have to find the number of numbers less than or equal to 2024 that are either congruent to 0 mod 5 or 2 mod 5. There are 404 numbers in the first category: 5, 10, 15, ..., 2020. For the second category, there are 405 numbers. 2, 7, 12, 17, ..., 2022. So the answer is 404 + 405 =