<p>Alice and Bob play the following game. A stack of <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> token or <img alt="$4$" height="12" src="https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png" width="9" /> tokens from the stack. Whoever removes the last token wins. Find the number of positive integers <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> less than or equal to <img alt="$2024$" height="13" src="https://latex.artofproblemsolving.com/e/5/7/e5711c8d31d3bc3119aba377c9bda0f9dfb14e4b.png" width="36" /> for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.</p>

Question No: 419

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.

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Tags: Combinatorics

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