Two knights endgame

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This article uses algebraic notation to describe chess moves.

The two knights ending is a chess endgame with a king and two knights versus a king, possibly with a pawn or two. With two knights versus a lone king, although there are checkmate positions, the superior side cannot force them against proper (and easy) defense. (The superior side can force stalemate.) Interestingly, if the lone king has a pawn (and sometimes with two pawns), then checkmate can be forced in some cases.

Contents 1 Two knights can't force checkmate 2 Troitzky line 2.1 Examples 3 Second Troitsky line 4 See also 5 References 6 External links

[edit] Two knights can't 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).

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.

The possible checkmate on the edge of the board is the basis of a problem by Alfred de Musset (see diagram). White checkmates in three moves with 1. Rd7 Nxd7 2. Nc6 N-any 3. Nf6# (Hooper and Whyld 1992).

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, et. al. 1993:11).

[edit] Troitzky line

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, ironically 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 the external link number 2 and Second Troitsky 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 Black's 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.

However, Anatoly Karpov lost an endgame with a pawn versus two knights to Veselin Topalov [1] although he had a theoretical draw with a pawn past the line; because of its rarity, Karpov seemed not to know the theory of drawing and headed for the wrong corner.[2]

John Nunn analyzed this endgame with an endgame tablebase and stated that "the analysis of Troitsky and others is astonishingly accurate" (Nunn 1995:265).

[edit] Examples

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 (see algebraic chess notation)

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.

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

[edit] Second Troitsky line

Since many of the wins when the pawn is blocked on or behind the Troitsky line require more than fifty moves (and thus would be draws under the fifty-move rule) Karsten Müller asked for the "second Troitsky 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. (See the external link below.)

[edit] See also

• Checkmate
• Endgame

[edit] References

•   Hooper, David and Whyld, Kenneth (1992). The Oxford Companion to Chess, 2nd Edition. Oxford University Press. ISBN 0-19-866164-9.  Reprint: (1996) ISBN 0-19-280049-3

•   Paul Keres (1974, 1984). Practical Chess Endings. Batsford. ISBN 0-7134-4210-7. 

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

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

[edit] 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.