In this position, 1. Ne7 or 1. Nh6 immediately stalemates Black. White can try instead:.
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Johann Berger gave this position, a draw with either side to move. With White to move:. 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 "Chess program Wilhelm".
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Archived from the original on December 8, In the position at right, White can try 1. Black can easily avoid this with, for example, This possible checkmate is the basis of some problems see below. In this position from a game  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 Black responded. A draw by threefold repetition could have been claimed on move 78 and at other times. 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 1 move , decreasing the material advantage and allowing the defending king to have a pawn can actually allow for a forced checkmate.
Another reason is that the pawn can block its own king's path without necessarily moving e. 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 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.
However, the checkmate procedure is difficult and long. In fact, it can require up to moves by White, so in competition often a draw by the fifty-move rule will occur first but see this article and Second Troitsky line section for the zone where the win can be forced within fifty moves. Troitsky showed that "on any placement of the black king, White undoubtedly wins only against black pawns standing on [the Troitsky line] and above" Rabinovich 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 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 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. But a fine win is in a game by Seitz , see Znosko-Borovsky vs.
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Seitz Giddins Two knights versus pawn is sometimes called the "Halley's Comet" endgame. 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. White to play draws. Black to play loses. 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.
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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 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 Anatoly Karpov lost an endgame with a pawn versus two knights to Veselin Topalov  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 Troitzky 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.
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