A few basic questions about combinatorial game theory

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SUMMARY

This discussion focuses on foundational concepts in combinatorial game theory, specifically regarding impartial games, the solution to Nim, and the Sprague-Grundy theorem. Key questions include the rationale behind using binary exclusive-or addition for computing Nim-sums and the distinction between Nim-sums and the Minimum Excludant (Mex) rule. Nim-sums are essential for determining winning and losing positions in Nim games, while the Mex rule serves a complementary purpose in analyzing game states. The conversation highlights the need for clearer explanations of these concepts and their interrelations.

PREREQUISITES
  • Understanding of combinatorial game theory fundamentals
  • Familiarity with impartial games
  • Knowledge of the Sprague-Grundy theorem
  • Basic concepts of binary arithmetic, specifically exclusive-or operations
NEXT STEPS
  • Research the application of the Sprague-Grundy theorem in various impartial games
  • Study the Minimum Excludant (Mex) rule and its role in combinatorial game theory
  • Explore advanced strategies for computing Nim-sums in complex game scenarios
  • Learn about the implications of binary operations in game theory analysis
USEFUL FOR

This discussion is beneficial for students and enthusiasts of combinatorial game theory, game theorists, and anyone interested in understanding the mathematical foundations of impartial games like Nim.

andrassy
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I am trying to teach myself the basic theory behind combinatorial game theory regarding impartial games, the solution to Nim, and the Sprague-Grundy theorem. I understand most of it, but I have a few questions about parts that are still unclear to me.
1. Why are Nim-sums computed using binary exclusiveor addition? I am having difficulty conceptualizing the theory behind using binary addition to represent nim heaps mathematically.
2. What is the difference/purpose of Nim-sums versus the Mex rule? I know that Nim-sums can be used to determine if any position in a game of Nim is winning or losing, and they they can also be used to determine the correct move. I don't really understand the mex rule and what it's point is. I have seen both of them but if someone could better explain mex and how it is related to Nim-sums and how it is used to solve Nim or impartial games in general, I would really appreciate it!
 
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If no one knows could someone maybe point me somewhere else I can ask?
 
Sorry for bumping this old thread, but I've got a new website that I just launched where you can go and ask me questions and learn about combinatorial game theory.

www.combinatorialgametheory.com

Doesn't have much there now because it's only a day old, but you can drop by and leave comments.
 
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