A few basic questions about combinatorial game theory

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The discussion focuses on understanding combinatorial game theory, specifically regarding impartial games, Nim, and the Sprague-Grundy theorem. Key questions include the rationale behind using binary exclusive OR (XOR) for calculating Nim-sums and the distinction between Nim-sums and the Minimum Excludant (Mex) rule. The user seeks clarification on how Nim-sums indicate winning or losing positions and the role of the Mex rule in this context. Additionally, a new website has been launched for further inquiries and learning about combinatorial game theory. The conversation emphasizes the need for clearer explanations of these concepts.
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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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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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