Tic-Tac-Toe Theory-An Escape From The Combinatorial Chaos

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Abstract: Tic-Tac-Toe Theory--an escape from the combinatorial chaosMathematics is spectacularly successful doing generalizations (Calculus, Modern Algebra, Algebraic Geometry, etc.); on the other hand, it could say surprisingly little about nontraditional complex systems like the economy or Chess. What is the fundamental problem with complex systems? The short answer is: the immense space of possibilities. The estimated total number of moves in Chess is more than ten-to-the-hundred, which is more than the number of particles in the observable universe. The majority of problems in Discrete Mathematics lead to cases studies of the same size. Statistical Mechanics faces the same problem: a mole of gas contains aboutten-to-23 particles; the underlying dynamic is incredibly complicated. The basic idea of Statistical Mechanics is to work with a priory probabilities; usually equiprobability in the phase space (it is counterintuitive: how does probability enter Newtonian mechanics?). The investigator develops aTic-Tac-Toe Theory which is a striking analog of the Statistical Mechanics approach: it can determine the exact value of infinitely many natural Game Numbers (the same paradox: how does probability enter Chess?). The investigator's book (700 pages) about the subject will come out in2007 at Cambridge University Press.Understanding complex systems is a major challenge for contemporary Mathematics (and Theoretical Computer Science, and Economy, etc.). Games are the perfect natural models. Also analyzing games is perhaps the best way to give the students a taste of mathematical discovery; it is the most natural mathematical experimentation. To integrate this new branch of game theory (the investigator's Tic-Tac-Toe Theory) into math education is an extremely promising aspect. Finally, understanding complex games may give unique insight into how human intelligence works.