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The two knights endgame is a chess endgame with a king and two knights versus a king. In contrast to a king and two bishops (on opposite-colored squares), or a bishop and a knight, a king and two knights cannot force checkmate against a lone king (however, the superior side can force stalemate [1] [2] ). Although there are checkmate positions, a king and two knights cannot force them against proper, relatively easy defense. [3]
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Paradoxically, although the king and two knights cannot force checkmate of the lone king, there are positions in which the king and two knights can force checkmate against a king and some additional material. [5] The extra material of the defending side provides moves that prevent the defending king from being stalemated [6] or, less commonly, the extra material obstructs the defending king from escaping check. The winning chances with two knights are insignificant except against a few pawns. [7] These positions were studied extensively by A. A. Troitsky, who discovered the Troitsky line, a line on or behind which the defending side's pawn must be securely blockaded for the attacking side to win.
If the side with the knights carelessly captures the other side's extra material, the game devolves to the basic two knights endgame, and the opportunity to force checkmate may be lost. When the defender has a single pawn, the technique (when it is possible) is to block the pawn with one knight, and use the king and the other knight to force the opposing king into a corner or nearby the blocking knight. Then, when the block on the pawn is removed, the knight that was used to block the pawn can be used to checkmate. [8]
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In general, two knights cannot force checkmate, but they can force stalemate. Three knights can force checkmate, [9] even if the defending king also has a knight [10] or a bishop. [11]
Edmar Mednis stated that this inability to force checkmate is "one of the great injustices of chess." [12]
Unlike some other theoretically drawn endgames, such as a rook and bishop versus rook, the defender has an easy task in all endings with two knights versus a lone king. Players simply have to avoid moving into a position in which the king can be checkmated on the next move, and there is always another move available in such situations. [13]
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The player with the lone king has to make a blunder to be checkmated. In this position, 1.Ne7 or 1.Nh6 immediately stalemates Black. White can try instead:
and now if Black moves 4...Kh8?? then 5.Nf7# is checkmate, but if Black moves
then White has made no progress. [14]
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Johann Berger gave this position, a draw with either side to move. With White to move:
and White has made no progress. With Black to move:
gives stalemate. [15]
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There are also checkmate positions with the inferior side's king on the edge of the board (instead of the corner), but again they cannot be forced. [16] In the position at right, White can try 1. Nb6+, hoping for 1...Kd8?? 2.Ne6#. Black can easily avoid this with, for example, 1... Kc7. This possible checkmate is the basis of some problems (see below).
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In this position from a 1949 game [17] between Pal Benko and David Bronstein, Black underpromoted to a knight. Black did not promote to a queen or any other piece because White could fork Black's king and his newly promoted piece (e.g. 104...f1=Q 105.Ne3+) immediately after the promotion.
White made the humorous move
forking Black's king and knight, but sacrificing the knight. Black responded
and a draw was agreed. [18] (A draw by threefold repetition could have been claimed on move 78 and at other times.)
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Another example is the eighth game of the 1981 World Chess Championship match between Anatoly Karpov and Viktor Korchnoi. [19] Black forces a draw by
Three knights and a king can force checkmate against a lone king within twenty moves (unless the defending king can win one of the knights). [21] Also, a complete computational retrograde analysis revealed that they can force checkmate only on the edge of the board. [22] [23]
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In some positions with two knights versus a pawn, the knights can force checkmate by gaining a tempo when the pawn has to move, or having the pawn obstruct its king from escaping check.
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1. Kh4 Kg2 2. Kg4 Kg1 3. Kh3 Kh1 4. Ng3+ Kg1 5. Nf3#
The pawn does not move; it assists mate by blocking the king's escape.
Even though two knights cannot force checkmate (with the help of their king) against a lone king (with the exception of positions where White wins in one move), decreasing the material advantage and allowing the defending king to have a pawn can actually allow for a forced checkmate. The reason that checkmate can be forced is that the pawn gives the defender a piece to move and deprives him of a stalemate defense. [25] Another reason is that the pawn can block its own king's path without necessarily moving (e.g. Kling & Horwitz position right).
The Troitsky line (or Troitsky position) is a key motif in chess endgame theory in the rare but theoretically interesting ending of two knights versus a pawn.
The line, assuming White has the two knights and Black the pawn, is shown left.
The Russian theoretician Troitsky made a detailed study of this endgame and discovered the following rule:
If the pawn is securely blockaded by a white knight no further down than the line, then Black loses, no matter where the kings are.
— Karsten Müller and Frank Lamprecht, Fundamental Chess Endings 2001
An example of the application of this rule is given in the diagram Müller and Lamprecht right; "... the position would be lost no matter where the kings are." [26]
However, the checkmate procedure is difficult and long. In fact, it can require up to 115 moves by White (assuming perfect play), [27] so in competition a draw by the fifty-move rule will occur first.
Troitsky showed that "on any placement of the black king, White undoubtedly wins only against black pawns standing on [the Troitsky line] and above". [28]
John Nunn analyzed the endgame of two knights versus a pawn with an endgame tablebase and stated that "the analysis of Troitsky and others is astonishingly accurate". [29] He undertook this checking after the very ending occurred in a critical variation of his post mortem analysis of a game he lost to Korchnoi in the 1980 Phillips and Drew Tournament in London. Neither player knew whether the position was a win for the player with the knights (Korchnoi).
Even when the position is a theoretical win, it is very complicated and difficult to play correctly. Even grandmasters fail to win it. Andor Lilienthal failed to win it twice in a six-year period, see Norman vs. Lilienthal and Smyslov vs. Lilienthal. But a fine win is in a game by Seitz, see Znosko-Borovsky vs. Seitz. [30]
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This diagram shows an example of how having the pawn makes things worse for Black (here Black's pawn is past the Troitsky line), by making Black have a move available instead of being stalemated.
If Black did not have the pawn move available, White could not force checkmate.
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The longest wins require 115 moves; this is one example starting with 1... Ne7. [31]
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This position is winnable, but the white pawn can be allowed to move only after 84 moves, making the win impossible under the fifty-move rule.
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In this study by André Chéron, White wins even though the pawn is well beyond the Troitsky line. [32]
Black to move is quicker. With White to move, he must maneuver to give the move to Black, as follows. 1.Kc3 Kb1 2.Kd2 Ka1 3.Kc1 Ka2 4.Kc2 (White then maneuvers to get the same position with vertical instead of horizontal opposition) 4...Ka1 5.Kb3 Kb1 6.Nb2 Kc1 7.Kc3 Kb1 8.Nd3 Ka1 9.Kc4 Ka2 10.Kb4 Ka1 11.Ka3 Kb1 12.Kb3 (Now White has enough time to bring the blockading N in to generate a mating net in time) 12...Ka1 13.Ne3 g2 14.Nc2+ Kb1 15.Na3+ Ka1 16.Nb4 g1=Q 17.Nbc2#
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White to play draws. Black to play loses. [33]
In the situation with Black's rook pawn blockaded on h3, if the black king can enter and remain in the area marked with crosses in the adjacent diagram, the game is a draw. Otherwise, White can force the black king into one of the corners not located in the drawing zone and deliver checkmate. Black cannot be checkmated in the a8-corner because the knight on h2 is too far away to help deliver mate: Black draws by pushing the pawn as soon as White moves the knight on h2. White to play in the diagram can try to prevent Black to enter the drawing zone with 1.Ke6, but Black then plays 1...Kg5 aiming to attack the knight on h2. White is compelled to stop this with 2.Ke5 which allows Black to return to the initial position with 2...Kg6, and White has made no progress. [34]
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Anatoly Karpov lost an endgame with a pawn versus two knights to Veselin Topalov [35] although he had a theoretical draw with a pawn past the Troitsky line; because of its rarity, Karpov seemed not to know the theory of drawing and headed for the wrong corner. (Depending on the position of the pawn, checkmate can be forced only in certain corners. [36] ) In this "rapid play" time control, the position in the game was initially a draw, but Karpov made a bad move which resulted in a lost position. Topalov later made a bad move, making the position a draw, but Karpov made another bad move, resulting in a lost position again. [37]
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This position from a blindfold game between Wang Yue and Viswanathan Anand leads to an example with a forced win even though the pawn is past the Troitsky line. [38] The game continued
blocking the pawn with the wrong piece. Black should have played 61...Ne4 62. c4 Nc5!, blocking the pawn on the Troitsky line with a knight, with a forced win. The game continued:
Black still has a theoretical forced win in this position, even after letting the pawn advance past the Troitsky line:
and Black has a forced checkmate in 58 more moves. [39] However, the actual game was drawn.
Fine & Benko, diagram 201
White to move wins in 96 moves. | Fine, ECE #1778
White to move wins in 87 moves. Lomonosov Tablebases
Black to move wins in 146 moves. |
Two knights can win in some cases when the defender has more than one pawn. First the knights should blockade the pawns and then capture all except one. The knights cannot set up an effective blockade against four connected pawns, so the position generally results in a draw. Five or more pawns usually win against two knights. [40]
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In this 1991 game between Paul Motwani and Ilya Gurevich, Black has blockaded the white pawns. In ten moves, Black won the pawn on d4. There were some inaccuracies on both sides, but White resigned on move 99. [41]
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There are positions of mutual zugzwang in the endgame with two knights versus one pawn. In this position, White to move draws but Black to move loses. With Black to move:
With White to move, Black draws with correct play. White cannot put Black in zugzwang:
and White has no way to force a win. [42]
The possible checkmate on the edge of the board is the basis of some composed chess problems, as well as variations of the checkmate with two knights against a pawn.
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In this problem by Alex Angos, White checkmates in four moves:
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A similar problem was composed by Johann Berger in 1890. The solution is:
followed by
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In this composition by Alfred de Musset, White checkmates on the edge of the board in three moves with:
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In this study composed by Sobolevsky, White wins by checkmating with two knights:
In this study composed by Ashot Nadanian, White wins by checkmating with two knights:
If 1...Re7, then 2.N6f5! Re1 3.Rxg6+ Kxh5 4.Rxh6+ Kg5 5.Nf3+ and White wins.
and checkmate on the next move, due to zugzwang; two white knights deliver four different checkmates: [49]
The first known composition where two knights win against one pawn is, according to Lafora, by Gioachino Greco in 1620. [50] In 1780, Chapais did a partial analysis of three positions with the pawn on f4 or h4. [51] In 1851 Horwitz and Kling published three positions where the knights win against one pawn and two positions where they win against two pawns. [52] The analysis by Chapais was revised by Guretsky-Cornitz and others, and it was included by Johann Berger in Theory and Practice of the Endgame, first published in 1891. However, the analysis by Guretsky-Cornitz was incorrect, and the original analysis by Chapais was, in principle, correct. [53] Troitsky started studying the endgame in the early 20th century and published his extensive analysis in 1937. [54] Modern computer analysis found it to be very accurate. [55]
Master games with this ending are rare — Troitsky knew of only six when he published his analysis in 1937. In the first four (from c. 1890 to 1913), the weaker side brought about the ending to obtain a draw from an opponent who did not know how to win. The first master game with a win was in 1931 when Adolf Seitz beat Eugene Znosko-Borovsky. [56] [57]
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The endgame is the final stage of a chess game which occurs after the middlegame. It begins when few pieces are left on the board.
Zugzwang is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move will worsen their position.
Stalemate is a situation in chess where the player whose turn it is to move is not in check and has no legal move. Stalemate results in a draw. During the endgame, stalemate is a resource that can enable the player with the inferior position to draw the game rather than lose. In more complex positions, stalemate is much rarer, usually taking the form of a swindle that succeeds only if the superior side is inattentive. Stalemate is also a common theme in endgame studies and other chess problems.
In chess, a smothered mate is a checkmate delivered by a knight in which the mated king is unable to move because it is completely surrounded by its own pieces, which a knight can jump over.
Checkmate is any game position in chess and other chess-like games in which a player's king is in check and there is no possible escape. Checkmating the opponent wins the game.
Triangulation is a tactic used in chess to put one's opponent in zugzwang. Triangulation is also called losing a tempo or losing a move.
The chess endgame with a king and a pawn versus a king is one of the most important and fundamental endgames, other than the basic checkmates. It is an important endgame for chess players to master, since most other endgames have the potential of reducing to this type of endgame via exchanges of pieces. Players need to be able to determine quickly whether a given position is a win or a draw, and to know the technique for playing it. The crux of this endgame is whether or not the pawn can be promoted, so checkmate can be forced.
In chess, a fortress is an endgame drawing technique in which the side behind in material sets up a zone of protection that the opponent cannot penetrate. This might involve keeping the enemy king out of one's position, or a safe zone the enemy cannot force one out of. An elementary fortress is a theoretically drawn position with reduced material in which a passive defense will maintain the draw.
In chess, the bishop and knight checkmate is the checkmate of a lone king by an opposing king, bishop, and knight. With the stronger side to move, checkmate can be forced in at most thirty-three moves from almost any starting position. Although this is classified as one of the four basic checkmates, it occurs in practice only approximately once in every 6,000 games.
The rook and pawn versus rook endgame is a fundamentally important, widely studied chess endgame. Precise play is usually required in these positions. With optimal play, some complicated wins require sixty moves to either checkmate, capture the defending rook, or successfully promote the pawn. In some cases, thirty-five moves are required to advance the pawn once.
In chess, opposition is a situation in which two kings are two squares apart on the same rank or file. Since kings cannot move adjacent to each other, each king prevents the other's advance, creating a mutual blockade. In this situation, the player not having to move is said to have the opposition. It is a special type of zugzwang and most often occurs in endgames with only kings and pawns. The side with the move may have to move their king away, potentially allowing the opposing king access to important squares. Taking the opposition is a means to an end, normally to force the opponent's king to move to a weaker position, and is not always the best thing to do.
In chess, a desperado is a piece that is either en prise or trapped, but captures an enemy piece before it is itself captured in order to compensate the loss a little, or is used as a sacrifice that will result in stalemate if it is captured. The former case can arise in a situation where both sides have hanging pieces, in which case these pieces are used to win material prior to being captured. A desperado in the latter case is usually a rook or a queen; such a piece is sometimes also called crazy or mad.
The chess endgame of a queen versus pawn is usually an easy win for the side with the queen. However, if the pawn has advanced to its seventh rank it has possibilities of reaching a draw, and there are some drawn positions with the pawn on the sixth rank. This endgame arises most often from a race of pawns to promote.
In chess, a swindle is a ruse by which a player in a losing position tricks their opponent and thereby achieves a win or draw instead of the expected loss. It may also refer more generally to obtaining a win or draw from a clearly losing position. I. A. Horowitz and Fred Reinfeld distinguish among "traps", "pitfalls", and "swindles". In their terminology, a "trap" refers to a situation where players go wrong through their own efforts. In a "pitfall", the beneficiary of the pitfall plays an active role, creating a situation where a plausible move by the opponent will turn out badly. A "swindle" is a pitfall adopted by a player who has a clearly lost game. Horowitz and Reinfeld observe that swindles, "though ignored in virtually all chess books", "play an enormously important role in over-the-board chess, and decide the fate of countless games".
The opposite-colored bishops endgame is a chess endgame in which each side has a single bishop and those bishops operate on opposite-colored squares. Without other pieces besides pawns and the kings, these endings are widely known for their tendency to result in a draw. These are the most difficult endings in which to convert a small material advantage to a win. With additional pieces, the stronger side has more chances to win, but still not as many as when bishops are on the same color.
A pawnless chess endgame is a chess endgame in which only a few pieces remain, and no pawns. The basic checkmates are types of pawnless endgames. Endgames without pawns do not occur very often in practice except for the basic checkmates of king and queen versus king, king and rook versus king, and queen versus rook. Other cases that occur occasionally are (1) a rook and minor piece versus a rook and (2) a rook versus a minor piece, especially if the minor piece is a bishop.
Much literature about chess endgames has been produced in the form of books and magazines. A bibliography of endgame books is below.
In a chess endgame of a king, bishop, and pawn versus a bare king, or, less commonly, a king, rook, and pawn versus a bishop and king, a wrong rook pawn is a rook pawn whose promotion square is the opposite color from the bishop's square color. Since a side's rook pawns promote on opposite-colored squares, and a bishop can only move on one color, one of the pawns may be the "wrong rook pawn". This situation is also known as having the wrong-colored bishop or wrong bishop. In many cases, the wrong rook pawn will only draw, when any other pawn would win. This is because the defending side can sometimes get their king to the corner in front of the pawn, after which the attacking side cannot chase the king away to enable promotion. A fairly common defensive tactic is to reach one of these drawn endgames, often through a sacrifice.
In a chess endgame, a wrong bishop is a bishop that would have been better placed on the opposite square color. This most commonly occurs with a bishop and one of its rook pawns, but it also occurs with a rook versus a bishop, a rook and one rook pawn versus a bishop, and possibly with a rook and one bishop pawn versus a bishop.
The queen and pawn versus queen endgame is a chess endgame in which both sides have a queen and one side has a pawn, which one tries to promote. It is very complicated and difficult to play. Cross-checks are often used as a device to win the game by forcing the exchange of queens. It is almost always a draw if the defending king is in front of the pawn.
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