Two knights endgame

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Troitzky line, two white knights can checkmate if the black pawn is blocked on or before the marked squaresThe 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 White's 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 [http://www.chesscafe.com/text/mueller36.pdf 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" harvcol|Nunn|1995|p=265.

Examples

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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.

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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

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Chéron, 1955
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White wins with either side to moveIf 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, wich was also analyzed by Troitsky. In this study by André Chéron, White wins even though the pawn is well beyond the Troitsky line harvcol|Müller|Lamprecht|2001|p=20.

Topalov-Karpov

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Topalov-Karpov, 2000
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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 [http://www.chessgames.com/perl/chessgame?gid=1295765 Topalov-Karpov] 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 harvcol|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. [http://www.chesscafe.com/text/mueller37.pdf Muller article]

econd Troitzky line

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Second Troitzky lineSince 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 "X"s, White can force a win within fifty moves more than 99 percent of the time. [http://www.chesscafe.com/text/mueller36.pdf The second Troitzky line]

Checkmate in problems

The possible checkmate on the edge of the board is the basis of some composed chess problems.

de Musset

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Alfred de Musset, 1849
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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# harvcol|Hooper|Whyld|1992.

obolevsky

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P. Sobolevsky, Shakhmaty v. SSSR, 1951
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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# harvcol|Nunn|1981|p=6.

ee also

* Checkmate
* chess endgame
* Chess endgame literature

Notes

References

* Citation
last = Dvoretsky| first = Mark| authorlink = Mark Dvoretsky
year = 2006
title = Dvoretsky's Endgame Manual
edition = second
publisher = Russell Enterprises
id=ISBN 1-888690-28-3

*Citation
last=Fine|first=Reuben|authorlink=Reuben Fine
year=1941
title=Basic Chess Endings
publisher=McKay
ID=ISBN 0-679-14002-6

* Citation
surname1=Hooper|given1=David|authorlink1=David Vincent Hooper
surname2=Whyld|given2=Kenneth|authorlink2=Kenneth Whyld
title=The Oxford Companion to Chess
year=1992
edition=second
publisher=Oxford University Press
ID=ISBN 0-19-866164-9 Reprint: (1996) ISBN 0-19-280049-3

* Citation
surname1=Keres|given1=Paul|authorlink1=Paul Keres
title=Practical Chess Endings
year=1984
publisher=Batsford
ID=ISBN 0-7134-4210-7
*Citation
surname1=Müller|given1=Karsten|authorlink1=Karsten Müller
surname2=Lamprecht|given2=Frank|authorlink2=Frank Lamprecht
year=2001
title=Fundamental Chess Endings
publisher=Gambit Publications
ID=ISBN 1-901983-53-6

*Citation
last=Nunn|first=John|authorlink=John Nunn
year=1981
title=Tactical Chess Endings
publisher=Batsford
ID=ISBN 0-7134-5937-9

*Citation
last=Nunn|first=John
year=1995
title=Secrets of Minor-Piece Endings
publisher=Batsford
ID=ISBN 0-8050-4228-8

*Citation
last=Seirawan|first=Yasser|authorlink=Yasser Seirawan
title=Winning Chess Endings
year=2003
publisher=Everyman Chess
ID=ISBN 1-85744-348-9

*citation
last=Troitzky|first=Alexey|authorlink=Alexey Troitsky
year=2006
title=Collection of Chess Studies (1937)
publisher=Ishi Press
ID=ISBN 0-923891-10-2 The last part (page 197-257) is a supplement of 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 called"The Damned Pawn" (in PDFs):
# [http://www.chesscafe.com/text/mueller35.pdf Part 1] about the Troitzky line and the technique
# [http://www.chesscafe.com/text/mueller36.pdf Part 2: the second Troitzky line solved] the winning line taking into account the 50-move rule, and more winning techniques and drawing zones.
* [http://www.chessvideos.tv/two-knights-vs-pawn.php Two Knights vs King and Pawn Simulator]

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Two knights endgame

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