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Zugzwang (German for "compulsion to move", IPA: [ˈtsuːk.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).
[edit] Zugzwang in chess
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| Black to move is in zugzwang, and loses (Flear 2004:11). |
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 the moving player's 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 (the white king gets to either the c5 or e5 square and wins one or more pawns and can advance his own pawn toward promotion). If it is White's move, there is no zugzwang (Flear 2004:11-12). 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 triangulation. Zugzwang is very common in king and pawn endgames, where it is frequently achieved through triangulation.
Pieces other than the king can also triangulate to achieve zugzwang — e.g., see the queen versus rook position at Philidor position. Zugzwang is a mainstay of chess compositions and occurs frequently in endgame studies.
[edit] Reciprocal zugzwang
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| Reciprocal zugzwang or mutual zugzwang: White to move draws, Black to move loses |
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.
[edit] Trébuchet
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| Trébuchet (extreme mutual zugzwang), whoever moves loses |
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).
Mark (or Marc) Bourzutschky has used computer analysis to find some complicated trébuchet positions. If White is on move, Black quickly drives White's king toward the corner and mates no later than move 8, e.g. 1.Kb2 (1.Nhg7 Qf4+ or 1.Nh4 Qe3+ also leaves White's king in trouble) Qg2+ 2.Kb3 Qb7+! 3.Ka3 Qb6 4.Nf4+ Kc4! 5.Ka2 Qb3+! 6.Ka1 Kb4 7.Ng7 Ka3 8.Nge6 Qb2#. Black on move must give ground, enabling White to gradually improve the positions of his pieces, e.g. 1...Kc4 (1...Kc3 allows 2.Nf2 Qxf2?? 3.Ne4+) 2.Kd2! Kd5 3.Ne3+ Ke5 4.Ng7 and White mates by move 42 according to Bourzutschky.-- scroll down to No. 282
[edit] Mined squares
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| Squares marked "1" are mined squares |
Mined squares are squares such that a player will fall into zugzwang if he moves onto the square. In the diagram on the right, if either king moves onto the square near it labeled "1", he falls into zugzwang if the other king moves into the mined square near him. These are a type of corresponding squares (Dvoretsky2003:18).
[edit] Zugzwang required to win
In some endgames, zugzwang is required to force a win. These include: rook (and king) versus king checkmate, two bishops versus king checkmate, bishop and knight versus king checkmate, queen versus rook, queen versus knight, queen versus two knights, and queen versus two bishops (Soltis 2003:79).
[edit] Zugzwang in the middlegame and complex endgames
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| Sämisch vs. Nimzowitsch, Copenhagen, 1923. White resigned. |
The game Fritz Sämisch – Aron Nimzowitsch, Copenhagen 1923,[1] 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 1.Rc1 or Rd1 (see algebraic chess notation) then 1...Re2 traps white's Queen; 1.Kh2 fails to 1...R5f3, also trapping the queen (white cannot play Bxf3 here because the bishop is pinned to the king); 1.g4 runs into …R5f3 2.Bxf3? Rh2 mate. 1.a3 is met by 1...a5 2.axb4 axb4 3.b3 Kh8 (waiting) 4.h4 Kg8 and White has run out of waiting moves and must lose material. Other white moves lose material in more obvious ways. Whether the position is true zugzwang is debatable, however, because 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 (Horowitz 1971:182).
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| Harper-Zuk, position after 36...Rf1: White is utterly helpless. |
Harper-Zuk, Halloween Open, Barnaby, British Columbia 1971[2] is a grisly example of zugzwang in the middlegame. White's queen, rook, knight, and king have a total of one legal move (Qh3), and that move loses the queen and then the game (... gxh3 followed by ... Qxg2#). The game concluded: 37.b5 Kh8 37...Nf5 and Nd4-e2 was crushing, but letting White self-destruct is even quicker. 38.a4 Kh7 39.a5 Kg8 0-1 After 40.axb6 axb6, white is forced to play 41.Qh3, and then it's mate in two: gxh3 42.Kh2 Qxg2#.
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| Van Dongen vs. Wijsman, Eindhoven 2005, position after White's 74th move. |
An unusual example of Zugzwang in a complicated endgame occurred in the position at right. On the previous move Black, with a winning position, had played 73...d4? and White responded 74.R(from d2)-d3!!, when Black, a knight up with three dangerous passed pawns, suddenly must fight for a draw. Tim Krabbé explains that the pawns on d4 and e4 are blocked and pinned, the knight is bound to the defense of e4, the rook is bound to the defense of d4, and the pawn on b4 is bound to the defense of the knight. Krabbé analyzes as best for Black 74...b3! 75.Rxd4 Rxd4 76.Rxc3 Rd8 77.Rxb3 Re8 78.Re3 Re5 79.Rc3 (79.Kxf6? Rxa5 82.Kg6 Ra1 83.f6 Rg1+ wins) Re8 80.Re3 Re5 81.Rc3 and the game will end in a draw by repetition of moves. Instead, Black played 74...Nb5? 75.Rxe4 Nd6 76.Re6 Rc6 77.Rxd4 Rxh6+ 78.Kxh6 Nxf5+ 79.Kg6 1-0[3]
[edit] See also
[edit] References
- Noam D. Elkies (1996). "On numbers and endgames: combinatorial game theory in chess endgames". Games of No Chance 29: 135-50.
[edit] External links
