Second Troitzky line, where knight and pawn could be located?

Question

I am trying to understand where the pieces should be located for the second Troitzky line (checkmate 2 knights vs a pawn in 50 moves limit) and fail to get it. So from wikipedia:

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 can be blocked on or behind one of the Xs, White can force a win within fifty moves more than 99 percent of the time.

The problem is that I still can not understand where the pawn and a blocking knight could be located.

Just to resolve ambiguity: I do not need to know how should white force a mate, I just need to understand which positions are possible to win with 50 moves rule. For example, are these positions a theoretical win?

[FEN "8/8/2p1N2k/8/2N5/5K2/8/8 w - - 0 1"]

,

[FEN "8/8/8/1N2pk2/1N6/8/8/4K3 w - - 0 1"]

,

[FEN "8/N7/6k1/7p/8/8/8/1K5N w - - 0 1"]

Answer 1

This has been already solved by Helmut Conrady, his elegant 15-page solution is available here.

It is filled with diagrams and examples as you wanted. Let me know if you have any questions regarding the diagrams or explanations given.

EDIT: In reply to the edited version of the question and the three positions added:

For the position to be winnable under 50 moves, there are two simple conditions:

• Either the black pawn is blocked by the knight on the 5th ranked
• Or the pawn is blocked on either b6/b7 squares or on g6/g7 squares.

This means:

• First position Winning for white in 50 moves because the pawn on c6 isn't advanced enough and is already blockaded, so no hope for black.
• Second position Draw because white cannot win it under the 50 moves limit, as no matter what he tries in the current position, the pawn will get the chance to go past the 5th rank. White isn't in time to block the pawn, two examples:

First:

[Title "pawn will advance to 4th"]
[StartFlipped "0"]
[FEN "8/8/8/1N2pk2/1N6/8/8/4K3 w - - 0 1"]

1.Nd6+ Kf4 2.Nd5+ Kf3 3.Nf5 e4

Second:

[Title "pawn will advance to 4th"]
[StartFlipped "0"]
[FEN "8/8/8/1N2pk2/1N6/8/8/4K3 w - - 0 1"]

1.Nd6+ Ke6 2.Ne4 Kf5 3.Ng3+ Kf4 4.Kf2 e4

• Third position: Draw: easy enough to see that white can't stop the immediate advance of the pawn past the 5th, so white cannot win under 50 moves.

In order to conclude, although if one of the conditions is fulfilled white will have a theoretical win under 50 moves, but that is only possible if no tempo is lost usually, so techniques like triangulation are continuously applied, as shown in the examples of the link.

Answer 2

My understanding is that for, say, the d5 dot, that would be a white knight on d4 and a black pawn on d5. The pawn could also be farther back, of course, and the knight can be on d5 or d6 if it is still blocking the pawn.

The same holds for the Xs but it doesn't quite guarantee a victory considering the 50-move rule.

Edited with responses to your diagrams:

[FEN "8/8/2p1N2k/8/2N5/5K2/8/8 w - - 0 1"]

White wins, no matter whose move it is. The pawn can be stopped at either c5 or c6, both of which are behind the second Troitzky line.

[FEN "8/8/8/1N2pk2/1N6/8/8/4K3 w - - 0 1"]

Draw, no matter whose move it is. The pawn will move to e4, which is past the second Troitzky line.

[FEN "8/N7/6k1/7p/8/8/8/1K5N w - - 0 1"]

Draw, no matter whose move it is. The pawn will move to h4, which is past the second Troitzky line.

Answer 3

The second Troitzky line rule as stated in the link given in the first answer (Dr. Müller's article) is in need of clarification.

Unless the phrase "securely blockaded" is taken to mean blockaded by a knight which is protected by the other knight then the rule must be wrong (see "Proposed changes to the Troitzky line article" section here).

The diagrams in the link appear to be just the second half of Dr. Müller's exposition on the ending, not illustrating the rule.

If "blockaded" is defined as in Wikipedia and as assumed in the second answer, then moves leading up to the blockade (or positions with a gap between pawn and blockading knight) are not taken into account by the rule, so the rule gives no answer in any of the positions shown. What is needed is the number of moves to mate on the squares rather than a dot (the moves being calculated on a DTM50 basis). This seems to invalidate both previous answers if they are based solely on the rule.

The rule doesn't say that White cannot win if the pawn passes the line, only not always. This also seems to invalidate both previous answers if they are based solely on the rule.

I believe there are now Syzygy DTM50 EGTBs which will give the answer in any given position. The first position (either side to play) is a win for white because it's under 50 moves with DTM play.

Play in this ending becomes horribly complicated when the 50 move rule is taken into account. A good exposition can be found here