Bob has a forced win if the starting position \( n \) is a P-position where the first player loses; these occur exactly when \( n \equiv 0 \) or \( 2 \pmod{5} \). Up to 2024, there are 809 such integers, so the answer is \(\boxed{809}\).
combinatorial game theory
winning and losing positions
modular pattern analysis
token removal game
Forced-losing positions (P) for first player occur when \( n \equiv 0 \) or \( 2 \pmod{5} \); Counting \( n \le 2024 \) with \( n \equiv 0,2 \pmod{5} \) gives \(\boxed{809}\).
Cost
$0
Input
0tokens
Output
0tokens
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