syzygy wrote:But would that not greatly diminish the significance of the second Troitzky line?
That would depend on what Helmut Conrady had in mind as the meaning of "securely blockaded".
I think Müller and Lamprecht Fundamental Chess Endings means exactly what I gave by "securely blockaded" in the case of the first Troitzky line, but whether that was Conrady's intention regarding the second is open to question. Certainly all exceptions to the first line rule are also exceptions to the second line rule with the same interpretation of "securely blockaded" being used in each. The positions I gave in
viewtopic.php?f=6&t=8145#p88303 are all actually schemes and represent rather a lot of different exceptions in total to interpretations of the phrase other than the one I used. One could take the view that the significance of a rule diminishes as the number of exceptions increases.
syzygy wrote:For the second Troitzky line (where the 50-move rule does count), this equivalence is gone.
Agreed (roughly speaking, because " the knight cannot easily be chased away by the enemy king" is not a lot clearer than "securely blockaded" - is the knight chased away by the enemy king in 8/8/8/8/7p/1K5N/5N2/1k6 w - - 0 1 for example?). Hence a specific meaning of "securely blockaded" is even more important in this case. In fact it seems the second Troitzky line rule is to some extent incorrect for any plausible meaning, irrespective of whether the dots are actually in the right place. (It's not obvious to me that it should even be a line).
syzygy wrote:When actually playing a game and reaching a KNNvKP position, the first goal for white seems to be to block the pawn with one of its knights. If that can be done within the second Troitzky line, you still don't know if the game can be won as the blockading knight is probably not (yet) guarded by the other knight.
I made the same point on the Wikipedia talk page some years ago. When I looked through recorded games I found that about half became knnkp with the pawn already blockaded, I think partly because these occur after a swap of White's last pawn, as one of a pair of blocked pawns, or maybe the knnkp ending is already being played before it is reached with one eye on the second Troitsky line. If this is not the case the second Troizky line rule seems to be pretty irrelevant. More relevant would be max DTM50 values for each of the squares.
By the way when I practice these endings I find it better to think of the blocking and guarding as a single process. I just do this as far back as possible without particularly thinking of the (first) Troitzky line. I say first because I just try to hit the DTM and reset the move counter as necessary (playing against Nalimov EGTBs). Of course if the pawn is blockaded and you already have control of the black king with the other pieces you don't go back to a "Troitzky line" position.
I put a couple of pgns in the same dropbox link mentioned in the previous reply to better explain. In draw_trap the pawn can be stopped on d4, but you can't play 1. Nd3 immediately, otherwise 1. ... Ke4 draws. I played 1. Kb4 to temporarily protect the d4 square. After Black's third move the other knight has to come in to give a Troitsky line position with mutually protecting knights. In Rybka_bad_hair Nc4, Kd2-c3 will blockade the pawn on c5 and protect the blockading knight, but this is no use because the black king can attack the corner knight instead. So I had to go for a blockade with the pawn on c3 instead (this is past the Troitsky line, but ok with the black king excluded from the a1 corner). A Troitsky line position with mutually protecting knights then comes on move 5. In the second example after say move 25, a Troitsky line position with mutually protecting knights
could be set up on the next move (so the position
is won) but you wouldn't want to actually do that.
syzygy wrote:Nothing beats a full KNNvKP tablebase with 50-move info.
Agreed. Especially not me.
That's why I'm asking for some easy positions (
viewtopic.php?f=6&t=8147) to get started.
Thanks for the technical info. I can see I'll have to get down to doing some programming again. I think the DTZ50 EGTBs will also be OK for practicing with the 50 move rule in effect.