Editing Two knights endgame (section)

===Troitsky line{{anchor|Troitzky line}}===
{{Chess diagram
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| The Troitsky line

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| Troitsky line positions when White has the two knights and Black the pawn
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| All possible positions given by Troitsky line for the black pawn}}
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| Müller and Lamprecht 2001

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| White wins no matter where the kings are.<ref>{{Harvcol|Müller|Lamprecht|2001}}</ref>
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{{Chess diagram small
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| Kling & Horwitz 1851.

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| White to play and win. 
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.
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Even though two knights cannot force [[checkmate]] (with the help of their [[king (chess)|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.<ref>{{Harvcol|Müller|Lamprecht|2001|pp=19–20}}</ref> 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 [[knight (chess)|knights]] versus a [[pawn (chess)|pawn]].

The line, assuming White has the two knights and Black the pawn, is shown left. 
 
The Russian theoretician [[A. A. Troitsky|Troitsky]] made a detailed study of this endgame and discovered the following rule: 
{{Quote|text=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.|sign=Karsten Müller and Frank Lamprecht|source=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."<ref>{{Harvcol|Müller|Lamprecht|2001}}</ref>

However, the checkmate procedure is difficult and long. In fact, it can require up to 115 moves by White (assuming perfect play),<ref>{{Harvcol|Müller|Lamprecht|2001|pp=19-20}}</ref> so in competition a [[draw (chess)|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".<ref>{{Harvcol|Rabinovich|2012|p=88}}</ref>

[[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".<ref>{{Harvcol|Nunn|1995|p=265}}</ref>
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 [[grandmaster (chess)|grandmasters]] fail to win it. [[Andor Lilienthal]] failed to win it twice in a six-year period, see [http://www.chessgames.com/perl/chessgame?gid=1090836 Norman vs. Lilienthal] and [http://www.chessgames.com/perl/chessgame?gid=1090903 Smyslov vs. Lilienthal]. But a fine win is in a game by [[Adolf Seitz|Seitz]], see [http://www.chessgames.com/perl/chessgame?gid=1151993 Znosko-Borovsky vs. Seitz].<ref>{{Harvcol|Giddins|2012|p=26}}</ref>
{{clear}}

====Examples====
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| White to move wins.
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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.

:'''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.
{{clear}}

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| Black to move wins in 115 moves.
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The longest wins require 115 moves; this is one example starting with '''1... Ne7'''.<ref>{{cite web|url=https://syzygy-tables.info/?fen=8/k7/8/4Kn2/8/P7/8/1n6_b_-_-_0_1 |title = Syzygy endgame tablebases}}</ref>
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| Black to move wins in 86 moves.
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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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====Pawn beyond the Troitsky line====
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| Chéron, 1955

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| White wins with either side to move.
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In this study by [[André Chéron (chess player)|André Chéron]], White wins even though the pawn is well beyond the Troitsky line.<ref>{{Harvcol|Müller|Lamprecht|2001|p=20}}</ref>

Black to move is quicker. With White to move, he must maneuver to [[Triangulation (chess)|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#

{{clear}}
{{Chess diagram small
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| Averbakh & Chekhover, position #251 
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| Drawing area marked with "×"
White to play draws. Black to play loses.<ref>{{Harvcol|Averbakh|Chekhover|1977|pp=119–120}}</ref>
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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.<ref>{{Harvcol|Averbakh|Chekhover|1977|pp=119–120}}</ref>
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====Topalov versus Karpov====
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| Topalov vs. Karpov, 2000

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| White won after 74. Ne2, even though the pawn was past the Troitsky line.
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[[Anatoly Karpov]] lost an endgame with a pawn versus two knights to [[Veselin Topalov]]<ref>[http://www.chessgames.com/perl/chessgame?gid=1295765 Topalov vs. Karpov]</ref> 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.<ref>{{Harvcol|Troitsky|2006}}</ref>) 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.<ref>[http://www.chesscafe.com/text/mueller37.pdf Müller article]</ref>
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====Wang versus Anand====
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| Wang vs. Anand, 2009

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| Position after 61. Kxa5
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This position from a [[blindfold chess|blindfold]] game between [[Wang Yue (chess player)|Wang Yue]] and [[Viswanathan Anand]] leads to an example with a forced win even though the pawn is past the Troitsky line.<ref>[http://www.chessgames.com/perl/chessgame?gid=1541026 Wang vs. Anand]</ref> The game continued
:'''61... Kc5''', 
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:
:'''62. c4 Ne4'''
:'''63. Ka4 Nd4'''
:'''64. Ka5'''.
Black still has a theoretical forced win in this position, even after letting the pawn advance past the Troitsky line:
:'''64... Nc6+'''
:'''65. Ka6 Kd6!!'''
:'''66. c5+ Kc7'''
and Black has a forced checkmate in 58 more moves.<ref>{{Harvcol|Soltis|2010|p=42}}</ref> However, the actual game was drawn.
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