Two knights endgame

This article uses algebraic notation to describe chess moves.

The two knights endgame is a chess endgame with a king and two knights versus a king, possibly with some pawns. In contrast to a king plus two bishops, or a bishop and a knight, a king and two knights can not force checkmate. (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).

Checkmate position, but it cannot be forced (Seirawan 2003:17). The knight on d2 could be on c3 or a3 instead and the white king could be on a3 instead.

Interestingly, if the lone king has a pawn (and sometimes with two 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).

Two knights cannot force checkmate

Two knights can not force checkmate

Although there are checkmate positions with two knights against a king, they cannot be forced. 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:

1. Nf8 Kg8

2. Nd7 Kh8

3. Nd6 Kg8

4. Nf6+

now if Black moves 4... Kh8?? then 5. Nf7# is checkmate, but if Black moves

4... Kf8

then White has made no progress (Keres 1984:2-3).

White can also try for a mate on the edge of the board

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).

Unlike some other theoretically drawn endgames, such as 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).

Example from game

Benko-Bronstein, 1949

Position before 106. Nh2+

In this position from a 1949 game^[1] between Pal Benko and David Bronstein, Black had just underpromoted to a knight. White made the humorous move

106. Nh2+

forking Black's king and knight, but sacrificing the knight. Black responded

106... Nxh2

and a draw was agreed (Benko 2007:133).

Troitzky line

Troitzky line, two white knights can checkmate if the black pawn is blocked on or before the marked squares

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]

Examples

White to move wins.

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.

1. Ne4 d2

2. Nf6+ Kh8

3. Ne7 (if Black did not have the pawn at this point, the game would be a draw because of stalemate.)

3. ... d1=Q

4. Ng6#.

If Black did not have the pawn move available, White could not force checkmate.

Black to move wins in 115 moves.

The longest win is this position that requires 115 moves, starting with 1... Ne7.

Pawn beyond the Troitsky line

Chéron, 1955

White wins with either side to move

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).

Topalov-Karpov

Topalov-Karpov, 2000

White won after 74. Ne2, even though the pawn was past the Troitzky line.

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]

Second Troitzky line

Second Troitzky line

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 within fifty moves. 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 is blocked on or behind one of the Xs, White can force a win within fifty moves more than 99 percent of the time.^[5]

More pawns

The 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 is a draw. Five or more pawns usually win against two knights (Fine & Benko 2003:101).

Fine & Benko, diagram 201

White to move wins

Fine, ECE #1778

White to move wins

Rinck, 1935, ECE #1780

White to move wins

Checkmate in problems

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.

Two knights versus one knight

Angos, 2005

White to move and mate in four moves

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:

1. Ne6! Nd8

2. Nf6+ Kh8

3. Ng5 N-any Black is in zugzwang and any knight move must abandon the protection of the f7 square.

4. Nf7# (Angos 2005:46)

Berger, 1890

White to move and mate in four moves

A similar problem was composed by Johann Berger in 1890. The solution is

1. Nf7! Nd6

2. Nh6+ Kh8

3. Ng5, followed by

4. Ngf7# (Matanović 1993:492-93).

de Musset

Alfred de Musset, 1849

White to checkmate in three moves.

In this composition by Alfred de Musset, White checkmates on the edge of the board in three moves with:

1. Rd7 Nxd7

2. Nc6 N-any

3. Nf6# (Hooper & Whyld 1992).

Sobolevsky

P. Sobolevsky, Shakhmaty v. SSSR, 1951

White to move and win.

In this problem composed by Sobolevsky, White wins by checkmating with two knights:

1. Nh8+ Kg8

2. Kxg2 Bf4

3. Ng6 Bh6!

4. Ng5 Bg7!

5. Ne7+ Kh8

6. Nf7+ Kh7

7. Bh4! Bf6!

8. Ng5+ Kh6

9. Ng8+ Kh5

10. Nxf6+! Kxh4

11. Nf3# (Nunn 1981:6).

History

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 published an extensive analysis in the early twentieth century, which modern computer analysis found 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).^[6]

Horwitz and Kling, 1851

White to move checkmates in eight moves

Horwitz and Kling, 1851

White to move wins

Pollock vs. Showalter, c. 1890

White to move. Pollock refused to continue and agreed to a draw six moves later but White has a winning position (Troitzky 2006:197).

See also

• Checkmate
• chess endgame

Notes

References

• Angos, Alex (2005), You Move ... I Win!: A Lesson in Zugzwang, Thinkers' Press, Inc., ISBN 978-1-888710-18-2 
• Dvoretsky, Mark (2006), Dvoretsky's Endgame Manual (2nd ed.), Russell Enterprises, ISBN 1-888690-28-3 

• Fine, Reuben (1941), Basic Chess Endings, McKay, ISBN 0-679-14002-6 
• Fine, Reuben; Benko, Pal (2003), Basic Chess Endings (1941) (second ed.), McKay, ISBN 0-8129-3493-8 
• Hooper, David; Whyld, Kenneth (1992), The Oxford Companion to Chess (second ed.), Oxford University Press, ISBN 0-19-866164-9  Reprint: (1996) ISBN 0-19-280049-3

• Horwitz, Bernhard; Kling, Josef (1986), Chess Studies and End-Games (1851, 1884), Olms, ISBN 3-283-00172-3 

• Keres, Paul (1984), Practical Chess Endings, Batsford, ISBN 0-7134-4210-7 
• Matanović, Aleksandar (1993), Encyclopedia of Chess Endings (minor pieces), 5, Chess Informant 
• Müller, Karsten; Lamprecht, Frank (2001), Fundamental Chess Endings, Gambit Publications, ISBN 1-901983-53-6 
• Nunn, John (1981), Tactical Chess Endings, Batsford, ISBN 0-7134-5937-9 

• Nunn, John (1995), Secrets of Minor-Piece Endings, Batsford, ISBN 0-8050-4228-8 

• Seirawan, Yasser (2003), Winning Chess Endings, Everyman Chess, ISBN 1-85744-348-9 

• Speelman, Jon; Tisdall, Jon; Wade, Bob (1993), Batsford Chess Endings, B. T. Batsford, ISBN 0-7134-4420-7 

• Troitzky, Alexey (2006), Collection of Chess Studies (1937), Ishi Press, ISBN 0-923891-10-2  The last part (pages 197–257) is a supplement containing Troitzky's analysis of two knights versus pawns.

External links

Grandmaster and endgame specialist Karsten Müller wrote a helpful two-part article on this endgame calledThe Damned Pawn (in PDFs):

1. Part 1 about the Troitzky line and the technique
2. Part 2: the second Troitzky line solved the winning line taking into account the 50-move rule, and more winning techniques and drawing zones.

• Two Knights vs King and Pawn Simulator