Zugzwang
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Zugzwang (German for "compulsion to move", IPA: [?tsuːk.tsvɑŋ]) is a term used in combinatorial game theory and in other types of games (particularly in chess). Zugzwang means that one player is put at a disadvantage because he or she has to make a move — the player would like to pass and make no move, but the fact that the player must make a move means being forced into a significantly weaker position. In combinatorial game theory, it means that it directly changes the outcome of the game from a win to a loss. The term is used less precisely in other games.
The term is frequently used in chess, but with a less precise meaning than in combinatorial game theory. In chess it is normally used to mean that one player (having the move) has no beneficial move (Soltis 2003:78). Game theory does not apply directly to chess (Berlekamp, et al, 1982:16; Elkies 1996:136). Sometimes different chess authors use the term zugzwang in different ways (Flear 2004:11-12). In a chess endgame, being in zugzwang usually means going from a drawn position to a loss or a won position to a draw, but it can be from a win to a loss, or a substantial loss of material which probably affects the outcome of the game. A chess position of reciprocal zugzwang or mutual zugzwang is equivalent to the more precise definition in game theory. Opposition is a special kind of zugzwang (Flear 2000:36).
Zugzwang in chess
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Normally in chess, having tempo is a good thing, since the player with the chance to move has greater power by being able to choose the "best" next move. Zugzwang typically occurs when all the moves available are "bad" moves, dramatically weakening his position (Müller and Lamprecht 2001:22).
Zugzwang most often occurs in the endgame when the number of pieces, and so the number of possible moves, is reduced, and the exact move chosen is often more critical. The first diagram gives a simple example. If it is Black's move, he gets to a lost position. If it is White's move, he is not in zugzwang, but he can use triangulation (king maneuvers on three adjacent squares in the shape of a triangle while the opposing king only has two squares) to return to the same position with Black to move. This puts Black in zugzwang. Zugzwang is extremely common in king and pawn endgames, where it is frequently achieved through triangulation.
Reciprocal zugzwang
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A special case of zugzwang is mutual zugzwang or reciprocal zugzwang, which is a position such that who ever is to move is in zugzwang. An example is shown in the second diagram — if White is to move the game is drawn; if Black is to move he loses (Flear 2004:22). According to John Nunn (Nunn 1999:7), positions of reciprocal zugzwang are surprisingly important in the analysis of endgames.
Trébuchet
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An extreme type of reciprocal zugzwang, called trébuchet is shown in the third diagram. It is also called a full-point mutual zugzwang because a full point (win versus loss) is at stake. Whoever is to move in this position loses the game — they must abandon their own pawn, thus allowing their opponent to capture it and proceed to promote their own pawn (Flear 2004:13).
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In May 2006 a trébuchet not involving knights or pawns was announced by Mark Bourzutschky (see diagram). White to move loses in two moves; Black to move loses in ninety-six moves.
Other pieces
Pieces other than the king can also triangulate — to achieve zugzwang — see for instance the queen versus rook position at Philidor position. Zugzwang is a mainstay of chess compositions and occurs frequently in endgame studies.Zugzwang required to win
In some endgames, zugzwang is required to force a win. These include: rook (and king) versus king, two bishops versus king, bishop and knight versus king, queen versus rook, queen versus knight, queen versus two knights, and queen versus two bishops (Soltis 2003:79).Zugzwang in the middlegame
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The game Fritz Sämisch – Aron Nimzowitsch, Copenhagen 1923, is sometimes called the "Immortal Zugzwang game" because the final position is widely accepted as being an extremely rare instance of zugzwang occurring in the middlegame. It ended with white resigning in the position in the diagram.
White has a few pawn moves which do not lose material, but eventually he will have to move one of his pieces. If he plays Rc1 or Rd1 (see algebraic chess notation) then …Re2 traps white's Queen; Kh2 fails to …R5f3, also trapping the queen (white cannot play Bxf3 here because the bishop is pinned to the king); g4 runs into …R5f3 Bxf3 Rh2 mate. Other white moves lose material in more obvious ways. Whether this is true zugzwang is debatable however, because even if white could pass his move he would still lose after …R5f3 Bxf3 Rxf3, when his queen is again trapped. (Horowitz 1971:182).
See also
- Opposition (chess)
- Null-move heuristic
- Seki
- Combinatorial game theory, in which all mutual zugzwangs are equivalent to 0.
- triangulation (chess)
- King and pawn versus king
References
- Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy (1982). Winning Ways for your Mathematical Plays (two volumes), Academic Press, ISBN 0-12-091101-9 (vol 1) ISBN 0-12-091102-7 (vol 2). (A second edition was published by A. K. Peters, in four volumes)
- Noam D. Elkies (1996). On numbers and endgames: combinatorial game theory in chess endgames, Games of No Chance, vol 29, 135-50.
- Glenn Flear (2000). Improve Your Endgame Play. Everyman Chess. ISBN 1-85744-246-6.
- Glenn Flear (2004). Starting Out: Pawn Endings. Everyman Chess. ISBN 1-85744-362-4.
- I. A. Horowitz (1971). All About Chess. Collier Books.
- Karsten Müller and Frank Lamprecht (2001). Fundamental Chess Endings, Gambit Publications. ISBN 1-901983-53-6
- John Nunn (1999). Secrets of Rook Endings. Gambit Publications. ISBN 1-901983-18-8
- Andrew Soltis (2003). Grandmaster Secrets: Endings, Thinkers' Press, ISBN 0-938650-66-1
External links
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