Bob has a forced winning strategy if and only if the initial number of tokens \( n \leq 2024 \) satisfies \( n \mod 5 = 0 \) or \( n \mod 5 = 2 \); there are 809 such integers.
combinatorial game theory
winning and losing positions
periodicity modulo
winning strategy count
Bob wins when \(n \equiv 0 \text{ or } 2 \pmod{5}\). Count \(n \leq 2024\) with \(n \equiv 0 \pmod{5}\): \(\lfloor 2024/5 \rfloor = 404\). Count \(n \leq 2024\) with \(n \equiv 2 \pmod{5}\): \(\lfloor (2024-2)/5 \rfloor + 1 = 404 + 1 = 405\). Total = \(404 + 405 = \boxed{809}\).
Cost
$0
Input
0tokens
Output
0tokens
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