The two knights endgame is a chess endgame with a king and two knights versus a king, possibly with some other material. The material with the defending king is usually one pawn, but some positions studied involve additional pawns or other pieces. In contrast to a king plus two bishops, 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.) Although there are checkmate positions, the superior side cannot force them against proper (and easy) defense (Speelman, Tisdall & Wade 1993:11).
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On the other hand, if the lone king has a pawn (and sometimes with more pawns), then checkmate can be forced in some cases. These positions were studied extensively by A. A. Troitzky. If the defender's pawn is blocked on or before the "Troitzky line", the stronger side can force checkmate, although it may require up to 115 moves with optimal play. 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 (Müller & Lamprecht 2001:19–20). The technique (when it is possible) is to block the pawn with one knight and use the king and other knight to force the opposing king into a corner or near the other knight. Then when the block on the pawn is removed, the knight can be used to checkmate (Dvoretsky 2006:280).
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Although there are checkmate positions with two knights against a king, they cannot be forced. Edmar Mednis stated that this inability to force checkmate is "one of the great injustices of chess" (Mednis 1996:40).
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. He simply has to avoid moving into a position in which he can be checkmated on the next move, and he always has another move available in such situations (Speelman, Tisdall & Wade 1993:11).
Incidentally, 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) (Fine 1941:5–6).
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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:
now if Black moves 4...Kh8?? then 5.Nf7# is checkmate, but if Black moves
then White has made no progress (Keres 1984:2–3).
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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 (Guliev 2003:74).
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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. 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[1] between Pal Benko and David Bronstein, Black had just underpromoted to a knight (104...f2–f1=N+ 105.Kd2–c3 Kg2–f3). Black did not promote to a queen because White could fork Black's king and queen 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 (Benko 2007:133).
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The Troitzky (or Troitsky) line (or Troitzky position) is a key motif in chess endgame theory in the rare and practically unimportant (but theoretically interesting) ending of two knights versus a pawn. The endgame was analyzed by A. A. Troitzky.
Whilst two knights cannot force checkmate (with the help of their king) against a lone king, a decrease in material advantage allowing the defending king to have a pawn can actually cause his demise. This is due to the fact that a common technique in this endgame is that of reducing the defending king to a position that would be a stalemate except for an available pawn move, and allowing the pawn to move can allow the attacking knights to move in for the kill. For the position with White on the attack, Troitsky established that if a black pawn is blockaded (by one of the white knights) on a square no further forward than the line a4–b6–c5–d4–e4–f5–g6–h4, then White can win the resulting endgame (and similarly in reverse for Black), no matter where the other pieces are placed. However, the checkmate procedure is difficult and long. In fact, it can require up to 115 moves by White, so in competition often a draw by the fifty-move rule will occur first (but see this article and Second Troitzky line section for the zone where the win can be forced within fifty moves). Therefore the ending is more of theoretical than practical interest. If the defending black pawn is past the Troitsky line, there are zones such that if the black king is in one, white still has a theoretical win; otherwise the position is a draw.
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" (Nunn 1995:265).
Two knights versus pawn is sometimes called the "Halley's Comet" endgame.[2]
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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 win is this position that requires 115 moves, starting with 1... Ne7 (this is not the only possible example of position where 115 moves are required to win).
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If a pawn is beyond the Troitsky line, the result usually depends on the location of the defending king. Usually there is a "drawing area" and a "losing area" for the defending king, which was also analyzed by Troitsky. In this study by André Chéron, White wins even though the pawn is well beyond the Troitsky line (Müller & Lamprecht 2001:20).
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In this diagram, if the black king can enter the drawing area and remain there, the game is a draw. Black cannot be checkmated in the a8 corner because the knight on h2 is too far away – the pawn would advance. Checkmate can be forced in the other two corners (Averbakh & Chekhover 1977:119).
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Anatoly Karpov lost an endgame with a pawn versus two knights to Veselin Topalov [3] although he had a theoretical draw with a pawn past the Troitzky 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 (Troitzky 2006).) 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.[4]
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This position from a blindfold game between Yue Wang and Vishy Anand is an example of a win with the pawn past the Troitsky line.[5] After
the pawn is blocked but it is past the Troitsky line – Black has a forced win (Soltis 2010:42). However, the actual game was drawn.
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Since many of the wins when the pawn is blocked on or behind the Troitzky line require more than fifty moves (and thus would be draws under the fifty-move rule) Karsten Müller asked for the "second Troitzky line", which corresponds to where the knights can win without the fifty-move rule coming into effect. If Black's pawn is blocked by a white knight on or behind one of the dots, White can force a win within fifty moves. If the pawn can be blocked on or behind one of the Xs, White can force a win within fifty moves more than 99 percent of the time.[6]
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 (Fine & Benko 2003:101).
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In this 1991 game between Paul Motwana 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 (Speelman, Tisdall & Wade 1993:114).
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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 (Averbakh & Chekhover 1977:106).
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 some positions it is possible for two knights to force checkmate against one knight, using the same idea, e.g. the defending knight makes a move available that would avoid stalemate. 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 problem composed by Sobolevsky, White wins by checkmating with two knights:
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In this problem 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:[8]
The fact that two knights can sometimes win against one or more pawns was known at least as early as 1780, when Chapsis did a partial analysis of three positions with the pawn on f4 or h4 (Troitzky 2006:200). In 1851 Horwitz and Kling published three positions where the knights win against one pawn and two positions where they win against two pawns (Horwitz & Kling 1986:64–68). The analysis by Chapsis was revised by Guretsky-Cornitz and others, and 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 Chapsis was in principle correct (Troitzky 2006:200). Troitsky started studying the endgame in the early 20th century and published his extensive analysis in 1937 (Mednis 1996:43). Modern computer analysis found it to be very accurate (Nunn 1995:265).
Master games with this ending are rare – Troitzky 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 (Troitzky 2006:197–99).[9]
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Grandmaster and endgame specialist Karsten Müller wrote a helpful two-part article on this endgame called The Damned Pawn (in PDFs):