A few basic questions about combinatorial game theory

In summary, the conversation discusses the topic of combinatorial game theory, specifically focusing on impartial games, the solution to Nim, and the Sprague-Grundy theorem. The individual has a few questions about how Nim-sums are computed and the difference between Nim-sums and the Mex rule. They are also seeking clarification on how the Mex rule is related to Nim-sums and how it is used to solve Nim or impartial games. They mention a new website where individuals can ask questions and learn more about combinatorial game theory.
  • #1
andrassy
45
0
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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  • #2
If no one knows could someone maybe point me somewhere else I can ask?
 
  • #3
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 [Broken]

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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What is combinatorial game theory?

Combinatorial game theory is a branch of mathematics that studies two-player games where outcomes are solely determined by the players' decisions and not by chance. It involves analyzing game strategies, predicting outcomes, and finding optimal moves for players.

What are the basic principles of combinatorial game theory?

The basic principles of combinatorial game theory include the use of mathematical tools such as game trees, game matrices, and combinatorial analysis to analyze games. It also involves studying the concept of game values, which represent the outcome of a game for a given player, and the concept of game equivalence, which determines if two games are essentially the same.

What types of games are studied in combinatorial game theory?

Combinatorial game theory primarily focuses on abstract games, such as chess, checkers, and Go, where there is no element of chance or hidden information. It also includes games with perfect information, where both players have complete knowledge of the game state, and games with imperfect information, where players have limited knowledge of the game state.

What are some real-world applications of combinatorial game theory?

Combinatorial game theory has applications in various fields, including computer science, economics, and artificial intelligence. It is used to analyze and design algorithms, optimize resource allocation, and develop intelligent decision-making systems.

How does combinatorial game theory differ from traditional game theory?

Combinatorial game theory differs from traditional game theory in that it focuses on games with discrete outcomes and no element of chance. Traditional game theory, on the other hand, includes games with continuous outcomes and elements of randomness. Combinatorial game theory also places more emphasis on the structure of the game and less on the players' preferences and motivations.

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