Related AI Model Resources

Explore more AI models, providers, and integration options:

  • Browse All AI Models
  • AI Providers Directory
  • MCP Servers
  • Integration Documentation
  • Pricing & Plans
  • AI Industry Blog
xai
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 \(\boxed{809}\).
about 1 month ago
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 \(\boxed{809}\).

xai/grok-3-beta

Cost
$0
Input
0tokens
Output
0tokens
Loading messages...

Traces

Start a Conversation or make an API call to see your traces appear here.

No span selected

Click on any span in the list to see detailed information