Wording of Troitsky line rule

Question

The current Wikipedia article on the two knights endgame states:

For the position with White to move, Troitsky established that if a black pawn is securely blockaded (by one of the white 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.

I think the wording needs to be changed because various positions seem to me to be exceptions to the rule as stated. I'm having difficulty persuading the author of this, so perhaps it's me misunderstanding it.

I thought, out of interest, that I might take a straw poll of how many people think that the draws below are adequately covered by the wording as it is and how many people think they area valid exceptions.

Please leave a comment saying "covered" or "exception" if you read this. (If you think "covered" it would be helpful to me if you could say why, but you don't need to add any explanation if you don't want to.)

I should mention here that, since this is a poll and I'm hoping for as many responses as possible, I won't actually accept any answers even if they are perfect.

(I removed my first diagram because it was "cooked" as pointed out by GloriaVictis. The diagram was as shown in his answer.)

2.

[fen "N6k/2p5/2N2K2/8/8/8/8/8 w - - 0 1"] 
[White "Either"]
[Black "to Move"]

3.

[fen "2N5/4p3/4N3/1k2K3/8/8/8/8 b - - 0 1"]
[White "Black"]
[Black "to Move"]

4.

[fen "7N/2p5/2N5/3K2k1/8/8/8/8 b - - 0 1"]
[White "Black"]
[Black "to Move"]

Answer 1

The key word in the rule is "securely".

Meaning, that there is nothing the opponent can do to lift the blockade.

In the posted position, white has to play 1. Ng6 - otherwise black will play 1. .. g5 and be beyond the Troitzky line - achieving draw immediately.

Subsequently, the black king will march toward the pawn, starting with 1. .. Kc5 and force the g6 knight away, as white is not in time to defend it, neither with the king nor the other knight. Then, g5 will and draw will follow.

Thus, the position appears to be covered by the rule in its current wording, as the pawn is not securely blockaded.

EDIT - answering to the comment.

4N3/6p1/8/8/1k5N/8/8/7K w - - 0 1

1. Ng6 Kc5 2. Kg2 (2. Nc7 Kd6 3. Nb5+ (3. Ne8+ {pointless}) Ke6 {white is one move short of getting the other knight to e5} 4. Nd4+ Kf6 {white must move the knight away and g5 follows}) Kd5 Kg3 Ke6 4. Kg4 (4. Nf4+ Ke7 {white has to move the knight away and g5 follows}) (4. Nc7+ Kf7 {white is not in time to defend the knight, g5 will inevitably follow after a couple checks}) Kf7 {white cannot defend both knights}

EDIT 2

As for the later added positions, I believe they are all covered by ETD's answer.

Answer 2

I believe that GloriaVictis' answer addresses your first position very well, in that the pawn is not actually securely blockaded in that case; Black can force the further advance of the pawn.

Your other positions are also susceptible to an insecure blockade, or are special in that, say, one of the knights is doomed to be lost, or else cannot participate in the play without capturing the pawn, yielding a draw in either case. For instance, position (4):

[fen "7N/2p5/2N5/3K2k1/8/8/8/8 b - - 0 1"]

1...Kf6 {The knight cannot be saved except by allowing the pawn to advance.} 2.Ne5 (2.Ke4 Kg7) c5 3.Nhf7

So, one could insist that the wording cited in your question be strengthened to say something like, "the pawn is securely blockaded and neither of the knights faces imminent capture, or is forced to capture the pawn in order to participate, etc.." Ultimately, that is perhaps a matter of taste; I think most would instead say that these are clearly recognizable as special cases which the wording you cite need not attempt to address.

Answer 3

For information I think the correct rule should be

If a black pawn is blockaded (by one of the white knights) on a square no further forward than the line a4–b6–c5–d4–e4–f5–g6–h4 and the blockading knight is protected by the other knight, then White can win the resulting endgame (and similarly in reverse for Black), no matter where the other pieces are placed.

Müller and Lamprecht imply this version in Fundamental Chess Endings but unfortunately don't make it explicit, which means that if the rules governing Wikipedia content are strictly adhered to, this also can't be made explicit on the Wikipedia page (failing an alternative source).