Zugzwang
Zugzwang (from
Although the term is used less precisely in games such as chess, it is used specifically in combinatorial game theory to denote a move that directly changes the outcome of the game from a win to a loss.[2][3] Putting the opponent in zugzwang is a common way to help the superior side win a game, and in some cases it is necessary in order to make the win possible.[4] More generally, the term can also be used to describe a situation where none of the available options lead to a good outcome.[5][6][7]
The term zugzwang was used in German chess literature in 1858 or earlier,[8] and the first known use of the term in English was by World Champion Emanuel Lasker in 1905.[9] The concept of zugzwang was known to chess players many centuries before the term was coined, appearing in an endgame study published in 1604 by Alessandro Salvio, one of the first writers on the game, and in shatranj studies dating back to the early 9th century, over 1000 years before the first known use of the term. International chess notation uses the symbol "⊙" to indicate a zugzwang position.
Positions with zugzwang occur fairly often in chess
Etymology
The word comes from German Zug 'move' + Zwang 'compulsion', so that Zugzwang means 'being forced to make a move'. Originally the term was used interchangeably with the term Zugpflicht 'obligation to make a move' as a general game rule. Games like chess and checkers have "zugzwang" (or "zugpflicht"): a player must always make a move on their turn even if this is to their disadvantage. Over time, the term became especially associated with chess.
According to chess historian Edward Winter, the term had been in use in German chess circles in the 19th century.[8]
Pages 353–358 of the September 1858 Deutsche Schachzeitung had an unsigned article "Zugzwang, Zugwahl und Privilegien". Friedrich Amelung employed the terms Zugzwang, Tempozwang and Tempozugzwang on pages 257–259 of the September 1896 issue of the same magazine. When a perceived example of zugzwang occurred in the third game of the 1896–97 world championship match between Steinitz and Lasker, after 34...Rg8, the Deutsche Schachzeitung (December 1896, page 368) reported that "White has died of zugzwang".
The earliest known use of the term zugzwang in English was on page 166 of the February 1905 issue of Lasker's Chess Magazine.[9] The term did not become common in English-language chess sources until the 1930s, after the publication of the English translation of Nimzowitsch's My System in 1929.[8]
History
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The concept of zugzwang, if not the term, must have been known to players for many centuries. Zugzwang is required to win the elementary (and common)
Other than basic checkmates, the earliest published use of zugzwang may be in this study by Zairab Katai, which was published sometime between 813 and 833, discussing shatranj. After
- 1. Re3 Ng1
- 2. Kf5 Kd4
- 3. Kf4
puts Black in zugzwang, since 3...Kc4 4.Kg3 (or Kg4) Kd4 5.Re1 and White wins.[14]
Polerio, 1585
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Philidor, 1777
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The concept of zugzwang is also seen in the 1585 endgame study by Giulio Cesare Polerio, published in 1604 by Alessandro Salvio, one of the earliest writers on the game.[15] The only way for White to win is 1.Ra1 Kxa1 2.Kc2, placing Black in zugzwang. The only legal move is 2...g5, whereupon White promotes a pawn first and then checkmates with 3.hxg5 h4 4.g6 h3 5.g7 h2 6.g8=Q h1=Q 7.Qg7#.[16]
Joseph Bertin refers to zugzwang in The Noble Game of Chess (1735), wherein he documents 19 rules about chess play. His 18th rule is: "To play well the latter end of a game, you must calculate who has the move, on which the game always depends."[17]
François-André Danican Philidor wrote in 1777 of the position illustrated that after White plays 36.Kc3, Black "is obliged to move his rook from his king, which gives you an opportunity of taking his rook by a double check [sic], or making him mate".[18] Lasker explicitly cited a mirror image of this position (White: king on f3, queen on h4; Black: king on g1, rook on g2) as an example of zugzwang in Lasker's Manual of Chess.[19] The British master George Walker analyzed a similar position in the same endgame, giving a maneuver (triangulation) that resulted in the superior side reaching the initial position, but now with the inferior side on move and in zugzwang. Walker wrote of the superior side's decisive move: "throwing the move upon Black, in the initial position, and thereby winning".[20]
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Paul Morphy is credited with composing the position illustrated "while still a young boy". After 1.Ra6, Black is in zugzwang and must allow mate on the next move with 1...bxa6 2.b7# or 1...B (moves) 2.Rxa7#.[21]
Zugzwang in chess
There are three types of chess positions: either none, one, or both of the players would be at a disadvantage if it were their turn to move. The great majority of positions are of the first type. In chess literature, most writers call positions of the second type zugzwang, and the third type reciprocal zugzwang or mutual zugzwang. Some writers call the second type a squeeze and the third type zugzwang.[22]
Normally in chess, having tempo is desirable because the player who is to move has the advantage of being able to choose a move that improves their situation. Zugzwang typically occurs when "the player to move cannot do anything without making an important concession".[23][24]
Hooper & Whyld 1992, p. 458
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Flear 2004, p. 11
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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 critical. The first diagram shows the simplest possible example of zugzwang. If it is White's move, they must either stalemate Black with 1.Kc6 or abandon the pawn, allowing 1...Kxc7 with a draw. If it is Black's move, the only legal move is 1...Kb7, which allows White to win with 2.Kd7 followed by queening the pawn on the next move.
The second diagram is another simple example. Black, on move, must allow White to play Kc5 or Ke5, when White wins one or more pawns and can advance their own pawn toward promotion. White, on move, must retreat their king, when Black is out of danger.[25] The squares d4 and d6 are corresponding squares. Whenever the white king is on d4 with White to move, the black king must be on d6 to prevent the advance of the white king.
In many cases, the player having the move can put the other player in zugzwang by using
Examples from games
Fischer vs. Taimanov, second match game
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Some zugzwang positions occurred in the second game of the 1971 candidates match between Bobby Fischer and Mark Taimanov.[26] In the position in the diagram, Black is in zugzwang because he would rather not move, but he must: a king move would lose the knight, while a knight move would allow the passed pawn to advance.[27] The game continued:
- 85... Nf3
- 86. h6 Ng5
- 87. Kg6
and Black is again in zugzwang. The game ended shortly (because the pawn will slip through and promote):[28]
- 87... Nf3
- 88. h7 Ne5+
- 89. Kf6 1–0
Fischer vs. Taimanov, fourth match game
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In the position shown, White has just gotten his king to a6, where it attacks the black pawn on b6, tying down the black king to defend it. White now needs to get his bishop to f7 or e8 to attack the pawn on g6. Play continued:
- 57... Nc8
- 58. Bd5 Ne7
- 59. Bc4! Nc6
- 60. Bf7 Ne7
Now the bishop is able to make a tempo move. It is able to move while still attacking the pawn on g6, and preventing the black king from moving to c6.
- 61. Be8
and Black is in zugzwang. Knights are unable to lose a tempo,[29] so moving the knight would allow the bishop to capture the kingside pawns. The black king must give way.
- 61... Kd8
- 62. Bxg6! Nxg6
- 63. Kxb6 Kd7
- 64. Kxc5
and White has a winning position. Either one of White's
Tseshkovsky vs. Flear, 1988
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This position from a 1988 game between Vitaly Tseshkovsky and Glenn Flear at Wijk aan Zee shows an instance of "zugzwang" where the obligation to move makes the defense more difficult, but it does not mean the loss of the game. A draw by agreement was reached eleven moves later.[34][35]
Reciprocal zugzwang
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A special case of zugzwang is reciprocal zugzwang or mutual zugzwang, which is a position such that whoever is to move is in zugzwang. Studying positions of reciprocal zugzwang is in the analysis of endgames.[10][11] A position of mutual zugzwang is closely related to a game with a Conway value of zero in game theory.[36]
In a position with reciprocal zugzwang, only the player to move is actually in zugzwang. However, the player who is not in zugzwang must play carefully because one inaccurate move can cause them to be put in zugzwang.[37] That is in contrast to regular zugzwang, because the superior side usually has a waiting move to put the opponent in zugzwang.[11]
The diagram shows a position of reciprocal zugzwang. If Black is to move, 1... Kd7 is forced, which loses because White will move 2. Kb7, promote the pawn, and win. If White is to move the result is a draw as White must either stalemate Black with 1. Kc6 or allow Black to capture the pawn. Since each side would be in zugzwang if it were their move, it is a reciprocal zugzwang.[38][39]
Trébuchet
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An extreme type of reciprocal zugzwang, called trébuchet, is shown in the diagram. It is also called a full-point mutual zugzwang because it will result in a loss for the player in zugzwang, resulting in a full point for the opponent.[40] Whoever is to move in this position must abandon their own pawn, thus allowing the opponent to capture it and proceed to promote their own pawn, resulting in an easily winnable position.[41]
Mined squares
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Corresponding squares are squares of mutual zugzwang. When there is only one pair of corresponding squares, they are called mined squares.[42] A player will fall into zugzwang if they move their king onto the square and their opponent is able to move onto the corresponding square. In the diagram here, if either king moves onto the square marked with the dot of the same color, it falls into zugzwang if the other king moves into the mined square near them.[43]
Zugzwang helps the defense
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Zugzwang usually works in favor of the stronger side, but sometimes it aids the defense. In this position based on a game between Zoltán Varga and Péter Ács, it saves the game for the defense:
- 1... Kc4!!
Reciprocal zugzwang.
- 2. Nc3 Kb4
Reciprocal zugzwang again.
- 3. Kd3 Bg7
Reciprocal zugzwang again.
- 4. Kc2 Bh6 5. Kd3 Bg7 6. Nd5+ Kxa4 7. Ke4 Kb5 8. Kf5 Kc5 9. Kg6 Bd4 10. Nf4 Kd6 11. h6 Ke7 12. h7 Bb2
This position is a draw and the players agreed to a draw a few moves later.[44]
Zugzwang in middlegames and complex endgames
Alex Angos notes that, "As the number of pieces on the board increases, the probability for zugzwang to occur decreases."
Sämisch vs. Nimzowitsch
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The game
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 1.Rc1 or Rd1, then 1...Re2 traps White's queen; 1.Kh2 fails to 1...R5f3, also trapping the queen, since White cannot play 2.Bxf3 because the bishop is pinned to the king; 1.g4 runs into 1...R5f3 2.Bxf3? Rh2 mate. Angos analyzes 1.a3 a5 2.axb4 axb4 3.h4 Kh8 (waiting) 4.b3 Kg8 and White has run out of waiting moves and must lose material. Best in this line is 5.Nc3!? bxc3 6.Bxc3, which just leaves Black with a serious positional advantage and an extra pawn.[49] Other moves lose material in more obvious ways.
However, since Black would win even without the zugzwang,[50] it is debatable whether the position is true zugzwang. Even if White could pass his move he would still lose, albeit more slowly, after 1...R5f3 2.Bxf3 Rxf3, trapping the queen and thus winning queen and bishop for two rooks.[51] Wolfgang Heidenfeld thus considers it a misnomer to call this a true zugzwang position.[52] See also Immortal Zugzwang Game § Objections to the sobriquet.
Steinitz vs. Lasker
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This game between
Podgaets vs. Dvoretsky
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Soltis writes that his "candidate for the ideal zugzwang game" is the following game
Fischer vs. Rossetto
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In this 1959 game[60] between future World Champion Bobby Fischer and Héctor Rossetto, 33.Bb3! puts Black in zugzwang.[61] If Black moves the king, White plays Rb8, winning a piece (...Rxc7 Rxf8); if Black moves the rook, 33...Ra8 or Re8, then not only does White gain a queen with 34.c8=Q+, but the black rook will also be lost after 35.Qxa8, 35.Qxe8 or 35.Rxe7+ (depending on Black's move); if Black moves the knight, Be6 will win Black's rook. That leaves only pawn moves, and they quickly run out.[62] The game concluded:
- 33... a5
- 34. a4 h6
- 35. h3 g5
- 36. g4 fxg4
- 37. hxg4 1–0[63]
Zugzwang Lite
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Jonathan Rowson coined the term Zugzwang Lite to describe a situation, sometimes arising in symmetrical opening variations, where White's "extra move" is a burden.[64] He cites as an example of this phenomenon in Hodgson versus Arkell at Newcastle 2001. The position diagrammed arose after 1. c4 c5 2. g3 g6 3. Bg2 Bg7 4. Nc3 Nc6 5. a3 a6 6. Rb1 Rb8 7. b4 cxb4 8. axb4 b5 9. cxb5 axb5 (see diagram). Here Rowson remarks,
Both sides want to push their d-pawn and play Bf4/...Bf5, but White has to go first so Black gets to play ...d5 before White can play d4. This doesn't matter much, but it already points to the challenge that White faces here; his most natural continuations allow Black to play the moves he wants to. I would therefore say that White is in 'Zugzwang Lite' and that he remains in this state for several moves.
The game continued 10. Nf3 d5 11. d4 Nf6 12. Bf4 Rb6 13. 0-0 Bf5 14. Rb3 0-0 15. Ne5 Ne4 16. h3 h5!? 17. Kh2. The position is still almost symmetrical, and White can find nothing useful to do with his extra move. Rowson whimsically suggests 17.h4!?, forcing Black to be the one to break the symmetry. 17... Re8! Rowson notes that this is a useful waiting move, covering e7, which needs protection in some lines, and possibly supporting an eventual ...e5 (as Black in fact played on his 22nd move). White cannot copy it, since after 18.Re1? Nxf2 Black would win a pawn. After 18. Be3?! Nxe5! 19. dxe5 Rc6! Black seized the initiative and went on to win in 14 more moves.
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Another instance of Zugzwang Lite occurred in
Zugzwang required to win
Soltis listed some endgames in which zugzwang is required to win:
- King and rook versus king
- King and two bishops versus king
- King, bishop, and knight versus king
- Queen versus rook
- Queen versus knight
- Queen versus two bishops
- Queen versus two knights.[67]
Positions where the stronger side can win in the ending of
See also
- Corresponding squares
- Forced move
- Key square
- Null-move heuristic
- Seki – a situation in Gowhere neither player can add a stone without disadvantage
Notes
- ^ There are several possibilities in the king and pawn versus king endgame. One is that the pawn can be captured before it promotes. Another one is that the pawn can promote without being blocked by the opposing king. The interesting cases are when the opposing king can block the pawn. In that case, whether or not the pawn can promote depends on the positions of the pieces and which side is to move. In the cases in which the pawn can be forced to promotion, zugzwang is required.
References
- ^ Soltis 2003a, p. 78
- ^ Berlekamp, Conway & Guy 1982, p. 16
- ^ Elkies 1996, p. 136
- ^ Müller & Pajeken 2008, pp. 173
- ^ Gabor, Daniela (8 September 2022). "Zugzwang central banking (ECB edition)". Financial Times. Archived from the original on 17 September 2022. Retrieved 8 August 2023.
Take the ECB, the posterchild for zugzwang central banking. It has four possible moves: raising rates, QT, holding rates and admitting regime defeat.
- PMID 34173570.
- ISSN 1879-3665.
Nevertheless, Russia has not yet entirely released itself from the zugzwang situation, as it has to further resort to reactionary acts to restrain a rising China, despite the fact that any move possible would further constrain Russia geopolitically stemming from Beijing's increasing power projection, chiefly in the economic domain (SREB).
- ^ a b c d Winter 1997
- ^ a b Winter 2008
- ^ a b Nunn 1995, p. 6
- ^ a b c Nunn 1999, p. 7
- ^ Soltis 2003a, p. 79
- ^ Davidson 1981, pp. 21–22, 41
- ^ Soltis 2009, p. 15
- ^ Angos 2005, pp. 108–9
- ^ Sukhin 2007, pp. 21, 23
- ^ Hooper & Whyld 1992, pp. 38–39
- ^ Philidor 2005, pp. 272–73
- ^ Lasker 1960, pp. 37–38
- ^ Walker 1846, p. 245
- ^ Shibut 2004, p. 297
- ^ Hooper 1970, pp. 196–97
- ^ van Perlo 2006, p. 479
- ^ Müller & Lamprecht 2001, p. 22
- ^ Flear 2004, pp. 11–12
- ^ Fischer vs. Taimanov 1971
- ^ Wade & O'Connell 1972, p. 413
- ^ Kasparov 2004, p. 385
- ^ Nunn 1995, p. 7
- ^ Silman 2007, pp. 516–17
- ^ Averbakh 1984, pp. 113–14
- ^ Flear 2007, pp. 286–87
- ^ Soltis 2003b, p. 246
- ^ Flear 2007, p. 241
- ^ Tseshkovsky vs. Flear, 1988
- ^ Stiller 1996, p. 175
- ^ Müller & Pajeken 2008, p. 179
- ^ Hooper 1970, p. 21
- ^ Averbakh 1993, p. 35
- ^ Nunn 2002, p. 4
- ^ Flear 2004, p. 13
- ^ Dvoretsky 2003, p. 87
- ^ Dvoretsky 2006, p. 19
- ^ Müller & Pajeken 2008, pp. 179–80
- ^ Angos 2005, p. 178
- ^ Angos 2005, p. 183
- ^ Sämisch vs. Nimzowitsch
- ^ Reinfeld 1958, p. 90
- ^ Angos 2005, p. 180
- ^ Nunn 1981, p. 86
- ^ Horowitz 1971, p. 182
- ^ Golombek 1977
- ^ "Steinitz vs. Lasker, World Championship Match 1896–97". Retrieved 2008-12-24.
- ^ a b Reinfeld & Fine 1965, p. 71
- ^ Whyld 1967
- ^ Soltis 2005, pp. 89–90
- ^ Soltis 2005, p. 90
- ^ Crouch 2000, pp. 36–37
- ^ Soltis 1978, pp. 55–56
- ^ Fischer vs. Rossetto
- ^ Soltis 2003b, p. 34
- ^ Giddins 2007, p. 108
- ^ Fischer 2008, p. 42
- ^ Rowson 2005, p. 245
- ^ a b Andrew Soltis, "Going Ape", Chess Life, February 2008, pp. 10–11.
- ^ "Portisch vs. Tal, Candidates Match 1965". ChessGames.com. Retrieved 2009-03-30.
- ^ Soltis 2003a, pp. 78–79
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Further reading
- ISBN 978-0-7134-7920-1
- Kaufman, Larry (September 2009), "Middlegame Zugzwang and a Previously Unknown Bobby Fischer Game", Chess Life, 2009 (9): 35–37