In this week’s Column, we shall have a look at the Troitzky Line in the endgame 2N vs. pawn.
The Troitzky Line
In all positions, White is the one who has the 2 knights whereas Black is defending with the pawn. The Troitzky Line indicates the last square the pawn must be stopped in order for White to be able to mate in all corners of the boards. Thus, White can still win if the pawn has crossed the line – if it is possible to trap the Black king in the right corner(s).
Let us take the pawns e-h in turn since the two sides of the board are symmetrical along the vertical middle line.
Pawn on h4 (rook’s pawn)
If the pawn is on h3 and therefore blocked on h2 by the reserve knight, it is not possible to mate in the a8-corner.
Winning against an h4-pawn with the Black king trapped in the a1-corner is by no means easy, as this earlier blog post by me shows. The win is there, but it is probably the most difficult endgame study ever composed (by Troitzky, of course).
Therefore, a4/h4 is on the Troitzky Line.
Pawn on g6 (knight’s pawn)
The essential question here is why the pawn is on g6 in the Troitzky Line, and not on g5? After all, it is possible to mate the king in all corners with a reserve knight on g4. The above diagram is a case in point:
1.Ne5! g4 2.Nd7 g3 3.Nb6+ Ka7 4.Nc8+ Ka8 5.Na4 g2 6.Nab6#
The reason is if the Black king is not trapped in the corner, and is able to harass the reserve knight on g4:
White to move plays 1.Ne4! Kg6 2.Ng3!+-. By covering the h5-square, the Black king is trapped and mated in the h8-corner.
Black to move plays 1…Kg6! 2.Ke6 Kh5 3.Kf5 Kh4=
White has no good way of kicking the Black king away from the reserve knight on g4, and thus no good way of making progress. If Black is allowed to push the pawn, the draw is clear for all to see.
Therefore, to cover all cases, the pawn is on b6/g6 in the Troitzky Line.
Pawn on f5 (bishop’s pawn)
The only reason the pawn is not on f4 (as in the diagram above), is the impossibility to mate under normal circumstances in the h1-corner as the g3-square is covered.
The above sequence by Troitzky shows that all the other 3 corners are possible, but that White needs to lose a tempo to win the opposition against Black’s king:
1.Kf8 Kf6 2.Ke8 Ke6 3.Kd8 Kd6 4.Kc8 Kc6 5.Kb8 Kb6 6.Ne4!
6…Kc6 7.Ka7 Kd5 8.Nf2! Kc6 9.Ka6+-
The king has been freed from the back rank.
Because the f4-pawn covers g3, the Troitzky Line has the pawn on f5 (and c5).
Pawn on e4 (central pawn)
With a pawn on e3 (blocked by a reserve knight on e2), mating in the a8-corner depends on who’s to move. The same goes for d3 vs. the h8-corner. The following study by Troitzky highlights this detail:
The solution without too many details goes like this:
1.Ne7+! Kg7 2.Rg1 Kh7 3.Rh1+ Kg7
The straightforward 4.Rxh8? Kxh8 5.Kf7 Kh7 6.Nf5 gives the position of mutual zugzwang later (see next diagram) with Black to move, and 6…Kh8!= is a draw.
4…Kg8 5.Rxh8+ Kxh8 6.Kf7! Kh7
A position of mutual zugzwang has arisen: with Black to move here, White would not be able to win. Fortunately, White is to move, and the release of the reserve knight leads to mate:
7.Ne4! d2 8.Nf6+ Kh8 9.Ne7 d1Q 10.Ng6#
The central pawns are on d4 and e4 in the Troitzky Line because of the mutual zugzwang between the d3-pawn and the h8-corner/the e3-pawn and the a8-corner.