Bob has a forced win if \(n \equiv 0 \text{ or } 2 \pmod{5}\); up to 2024, there are 809 such integers.
combinatorial game theory
winning and losing positions
modulo 5 pattern
token removal game
Bob winning counts for \(n \leq 2024\) with moves \(\{1,4\}\) **Losing (P) positions:** \(n \equiv 0 \text{ or } 2 \pmod{5}\) Count: multiples of 5 up to 2024 = 404 Count: numbers \(\equiv 2 \pmod{5}\) up to 2024 = 405 **Total:** \(404 + 405 = 809\)
Cost
$0
Input
0tokens
Output
0tokens
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