Combinatorial Games
Tic-Tac-Toe Theory
By József Beck
Publisher: Cambridge University Press
Print Publication Year: 2008
Online Publication Date:July 2010
Online ISBN:9780511735202
Hardback ISBN:9780521461009
Paperback ISBN:9780521184755
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511735202.004
Subjects: Discrete Mathematics, Information Theory and Coding
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Chess, Tic-Tac-Toe, and Hex are among the most well-known games of complete information with no chance move. What is common in these apparently very different games? In either game the player that wins is the one who achieves a “winning configuration” first. A “winning configuration” in Tic-Tac-Toe is a “3-in-a-row,” in Hex it is a “connecting chain of hexagons,” and in Chess it is a “capture of the opponent's King” (called a checkmate).
The objective of other well-known games of complete information like Checkers and Go is more complicated. In Checkers the goal is to be the first player either to capture all of the opponent's pieces (checkers) or to build a position where the opponent cannot make a move. The capture of a single piece (jumping over) is a “mini-win configuration,” and, similarly, an arrangement where the opponent cannot make a move is a “winning configuration.”
In Go the goal is to capture as many stones of the opponent as possible (“capturing” means to “surround a set of opponent's stones by a connected set”).
These games are clearly very different, but the basic question is always the same: “Which player can achieve a winning configuration first?”.
The bad news is that no one knows how to achieve a winning configuration first, except by exhaustive case study. There is no general theorem whatsoever answering the question of how. The well-known strategy stealing argument gives a partial answer to when, but doesn't say a word about how.
pp. i-vi
pp. vii-x
pp. xi-xiv
A summary of the book in a nutshell : Read PDF
pp. 1-14
PART A - WEAK WIN AND STRONG DRAW : Read PDF
pp. 15-16
Chapter I - Win vs. Weak Win : Read PDF
pp. 17-90
Chapter II - The main result: exact solutions for infinite classes of games : Read PDF
pp. 91-192
PART B - BASIC POTENTIAL TECHNIQUE – GAME-THEORETIC FIRST AND SECOND MOMENTS : Read PDF
pp. 193-194
Chapter III - Simple applications : Read PDF
pp. 195-229
Chapter IV - Games and randomness : Read PDF
pp. 230-304
PART C - ADVANCED WEAK WIN – GAME-THEORETIC HIGHER MOMENT : Read PDF
pp. 305-306
Chapter V - Self-improving potentials : Read PDF
pp. 307-379
Chapter VI - What is the Biased Meta-Conjecture, and why is it so difficult? : Read PDF
pp. 380-458
PART D - ADVANCED STRONG DRAW – GAME-THEORETIC INDEPENDENCE : Read PDF
pp. 459-460
Chapter VII - BigGame–SmallGame Decomposition : Read PDF
pp. 461-503
Chapter VIII - Advanced decomposition : Read PDF
pp. 504-551
Chapter IX - Game-theoretic lattice-numbers : Read PDF
pp. 552-609
Chapter X - Conclusion : Read PDF
pp. 610-657
Appendix A - Ramsey Numbers : Read PDF
pp. 658-668
Appendix B - Hales–Jewett Theorem: Shelah's proof : Read PDF
pp. 669-676
Appendix C - A formal treatment of Positional Games : Read PDF
pp. 677-704
Appendix D - An informal introduction to game theory : Read PDF
pp. 705-715
Complete list of the Open Problems : Read PDF
pp. 716-723
What kinds of games? A dictionary : Read PDF
pp. 724-726
Dictionary of the phrases and concepts : Read PDF
pp. 727-729
pp. 730-732