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On Numbers and Endgames: Combinatorial Game Theory in Chess Endgames

By Elkies, Noam D.

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Book Id: WPLBN0000674364
Format Type: PDF eBook
File Size: 240.07 KB
Reproduction Date: 2005

Title: On Numbers and Endgames: Combinatorial Game Theory in Chess Endgames  
Author: Elkies, Noam D.
Language: English
Subject: Science., Mathematics, Logic
Publication Date:


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Elkies, N. D. (n.d.). On Numbers and Endgames: Combinatorial Game Theory in Chess Endgames. Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: It was already noted in Winning Ways [Berlekamp et al. 1982, p. 16] that combinatorial game theory (CGT) does not apply directly to chess, because the winner of a chess game is in general not determined by who makes the last move, and indeed a game may be neither won nor lost at all but drawn by infinite play.1 Still, CGT has been effectively applied to other games such as Dots-and-Boxes and Go, which are not combinatorial games in the sense ofWinning Ways. The main difficulty with doing the same for chess is that the 8 8 chessboard is too small to decompose into many independent subgames, or rather that some of the chess pieces are so powerful and influence such a large fraction of the board's area that even a decomposition into two weakly interacting subgames (say a kingside attack and a queenside counteroensive) generally breaks down in a few moves. Another problem is that CGT works best with cold games, where having the move is a liability or at most an in nitesimal boon, whereas the vast majority of chess positions are hot: Zugzwang2 positions (where one side loses or draws but would have done better if allowed to pass the move) are already unusual, and positions of mutual Zugzwang, where neither side has a good or even neutral move, are much rarer...


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