White to play and mate in 21.

There are two keys (essentially the same).

I recently tried playing the White pieces from this position against the Nalimov database but stuck at mate in 30 and had to consult my opponent for the solution (which turned out to be surprising).

The number of moves here is as much part of of the problem as the key moves.

In the first version I use suboptimal Black defence to better illustrate the idea of the puzzle. This is actually mate in 16. The second uses optimal Black defense and you have to avoid falling into the the same trap that I did. 

(I don't know how you handle multiple lines here, so expect many incorrect "incorrect"s including one of the keys, but if you've found the "incorrect" key the "correct" key will be obvious.)

The Troitsky line is not an absolute. Often it depends on which part of the board the black king is. Here it is apparently the 'wrong part'.

In fact, here the pawn is well beyond the Troitsky line.

White to play and win

I've been aware of draws beyond the Troitsky line for a long time (see also diagram), but until I tried this position, I'd always assumed that the farther back the pawn could be stopped the quicker would be the mate.

What I like about this (the originally posted) position is that not only is it an exception to that assumption, but also an exception to an exception, in that almost all positions with the knights and pawn in the positions shown are drawn and therefore exceptions to the "Troitsky line rule" as normally stated, but this is one of four exceptions to this exception where the rule holds good.

(Personally I doubt that Troitsky ever stated the "Troitsky line rule", though he no doubt came up with the Troitsky line itself in a related context. Certainly it's not completely valid with any reasonable wording.)

"Personally I doubt that Troitsky ever stated the "Troitsky line rule", though he no doubt came up with the Troitsky line itself in a related context. Certainly it's not completely valid with any reasonable wording."

According to Müller and Lamprecht, Troitsky discovered the following rule:

If the pawn is securely blockaded by a white knight no further down than the line, then Black loses, no matter where the kings are.

In position #1, this rule does not predict a win for White, since the pawn, although blockaded, is not "securely blockaded" and must be released for White to have any winning chances. The wins with the pawn beyond the line are also not exceptions, since the rule does not address those cases.

Would you consider the above rule to be reasonably worded? What about its validity?

I think it is fairly worded. I expect however that the knights side in an actual game will attempt to win wherever the blockade occurs. The important thing to know is that the blockade must be maintained and have an idea on how to corner the opposing king. Troitsky may be looking on in pain, but it can hurt no-one to try for a while (except the pawn side).

In all (but a handful of) cases White must release the blockade in order to win.

To me "securely blockaded" means Black cannot force the release of the blockade, i.e. the blockade is "secure" so long as White does not cede the win (or draw as the case may be) or increase the number of moves he needs to win or decrease the number of moves Black needs to win, by keeping it in place. This may change with play (and will if White has a win). 

With this interpretation of "securely" then certainly the antecedent to Troitsky line rule as stated by M&L is false in #1 because White must immediately release the blockade or cede a half point.

In any case I would hardly claim that #1 is an exception to either M&L's version of the rule or any of numerous other versions that appear, because the consequent (i.e. that White can win) is in this case true, so the rule holds whether the antecedent is judged true or false. Similarly I would agree (and I think have everywhere agreed) with you that if the pawn is beyond the line these cannot be exceptions whatever the result because the antecedent is false.

The M&L statement is, I would say, well worded, although it might benefit from some clarification of the words "securely" and "blockaded" which people take in different ways.

I recently replaced the version that appeared in Wkipedia with the M&L version. I was in correspondence with Dr. Müller, who said that the words, "no matter where the kings are"  (my italics) had been included by the authors specifically to cope with the exceptions to the rule and I could replace the word "securely" by saying the blockaded knight was guarded by the other knight. However since Wikipedia policy disallows personal correspondence as a source the replacement was never made.

As for validity the M&L statement has for fewer exceptions than other versions (so long as it is taken as suggested in the previous paragraph). If the word "blockaded" is taken to mean that the knight stands directly in front of the pawn then there are exactly 36 (legal) exceptions (including reflections across the board) which essentially boil down to just three cases. With other statements the number of exceptions ranges between tens of thousands and many millions. (In cases where these include the word "securely" this assumes the meaning I gave above - these generally do not come with their own definition.) If the word "blockaded" is taken to mean the knight stands on the pawn's file somewhere in front of the pawn then there would be a few thousand exceptions to M&L's version; this interpretation increases the number of exceptions in most versions of the rule.

While M&L and many others  say that Troitsky discovered the rule none that I can find gives a reference to where where he states it. It does not occur in his 60 page supplement to "Collection of Chess Studies" which deals exclusively with the subject, nor does the material support the rule. I remain sceptical that Troitsky originated the rule.  

By the way there is another rule associated with the Troitsky line that I believe Troitsky did state, namely:

The black king has a nonempty win/draw zone for a particular position of the pawn if and only if that position is not on or behind the Troitsky line.

This includes nothing about blockades ("secure" or otherwise). A win/draw zone is a set of squares associated with the pawn position which, if occupied by the black king ensures either a Black win or a draw irrespective of the positions of the White pieces.

This is completely accurate.

This of course does not preclude Black wins or draws with the black king outside the zone for particular positions of the White pieces whether the pawn be behind or in front of the line.

I am not sure what you are saying but it is clear to me that the Troitsky line must be associated with a blockaded pawn. Otherwise the pawn would just keep on pushing until it is over that line (as far as it can).

And obviously that blockade must be lifted at some point, since you can't mate the king with just one knight. But that's when we entered the phase of the checkmate count-down.

Do you disagree with any part of this?

The OP showed an interesting position (#5) where both sides get to win by moving first. I thought I would show one (unrelated to Troitsky) where both sides get to lose by moving first.

White to play loses, Black to play loses

Wow, precisely as in my last game against Robert J. Fisher! I don't remember which side I was on, but unfortunately, it was my move.

Actually, I was hoping it might inspire you to compose another pawnless full-point reciprocal zugzwang position.

Not easy, not easy at all. To a composer, reciprocals are not an end in themselves since there is always one side actually on move. The reciprocal only pays off when there is a try line with the same position in the other tempo. A challenge might be to take your diagram and turn it into a actual study. Waiting for inspiration .... waiting .... waiting .... waiting ....

Well, I haven't disagreed with anybody so far in these discussions but, since I am asked, I must pick up on some of the points you make here.

But first; the Troitsky line is associated with the rule I gave in the previous post and no it does not refer to blockaded pawns. I believe Troitsky enumerated the squares of the "Troitsky line" (without christening it) in his enunciation of the rule. This is not the rule commonly referred to as "the Troitsky line rule".  The win/draw zones referred to are zones where Black is guaranteed a win or draw irrespective of the positions of the White pieces. In some of these White may well be unable to block or capture the pawn on or before its promotion square and Black may (or may not) then have a won queen versus two knights endgame.

Hopefully an example will clarify what I said about zones. In the following diagram, there are no forced White win positions with either side to move if the black pawn is on the square shown and the black king is on the square shown or any one of the squares with a queen shown. This applies wherever the White pieces are placed. The black pawn may or may not be blockaded in particular placements (though of course it will be in the cases where the black king itself is blockading - either those positions on the b file or just b4 depending on your understanding of the term "blockaded").

The area covered by the king/queens is the Black kings win/draw zone for that placement of the pawn. If the black king is in the zone, Black has either a win or a draw whatever the position of the White pieces. If he is outside it then White wins with some possible placements of the White pieces, but there will also be individual Black wins and draws.

Notice that the pawn is beyond the Troitsky line in the example shown. On or behind it Black's win/draw zone disappears. Again this does not preclude individual Black wins and draws.

The Troitsky line is also associated with the rule(s) usually referred to as "the Troitsky line rule" In this case the rule does relate to blockaded pawns In most statements blockaded by a White knight.

You say that the blockade need be lifted only at the "checkmate count down". 

This isn't true. A standard technique is to lift the blockade while keeping a knight handy to reinstate it a square (or more) further down, and many positions (including all but a trivial number of rook's pawn positions where the pawn has not reached the sixth rank) cannot be won without using it.

Indeed the "checkmate count down" needn't necessarily exist. In the "mates in 11" diagram shown above, the pawn must not be allowed to promote. Its job is to stay put and assist in the mate by blocking its king.

You also say that you can't mate a king with just one knight. True, but you can mate a king with just one knight and one (or more) of his pawns. The rooks pawn variants of this ending rely on this fact.

Hope that clarifies.

I am not in the possession of historical records to prove you right or wrong but I am pretty sure that chess players refer to the Troitsky line as the familiar 8-pawn pattern. Irrespective of what Troitsky precisely said or wrote, he likely was not the name-giver to his own line. That feat was performed by others as a (possibly simplified) interpretation of his work. And it involves blockaded pawns and not zones.

I have played quite a number of these endgames against a tablebase engine and it is not common to lift a blockade before the mating phase, if that pawn is already on (over) the Troitsky line. Tablebases sometimes lift it before we would since they go for the fastest mate and not the one humans understand. But speed is something else than principle.

Of course, there is the little issue of what the "mating phase" exactly constitutes. Roughly I would say the last 8 moves or the black king being locked in his final corner.

Note that these principles fully apply to your post #1. Once the pawn is blockaded, it remains so until 4 moves before the mate. I never stated that the blockade had to start before (or on) the Troitsky line.

How important is the Troitsky line in practical play? Since the position is usually won if the pawn is not past the line, and possibly won if it is, I would say it makes sense for the strong side to play on for a while in any case.

Knowing about the line may be more beneficial to problemists, since it lets them say things like "Look at this interesting position, where the knights win even though the pawn is well past the Troitsky line."

I first learned how to play the ending from Averbakh's "Knight Endings". He doesn't even mention the Troitsky line.

I have played a ludicrous number of these positions against the Nalimov database and I never think in terms of the Troitsky line. If I'm playing the two knights, I just first ask the basic questions, "Can I mate without blocking the pawn?"  and "Can I stop the pawn on or before its queening square?". If the answer to both is "no" then it becomes interesting. Otherwise, if the answer to only the first is "no" I go for a blockade on the first square I can make it stick, with a knight if I have a choice. But this puzzle shows that is not always correct.

In any case I would say "the Troitsky line rule" is pretty irrelevant. But the second rule I gave above is relevant The same rule applies to pawns blockaded by a knight (but with different zones) . It is important White not to let the Black king into these zones if he is to win. If Black can't get his pawn home and free then he can also attempt to reach the relevant zone. So I think the Troitsky line itself is relevant.

Of course if you have practised a lot you just know where you shouldn't allow the king or respectively where you should try to get to. It just so happens that you don't have a problem or you have nowhere to go exactly when the pawn hasn't passed the Troitsky line. You can still play it perfectly well without ever having heard of line. 

Interestingly (and incomprehensibly) the whole of the Wikipedia treatment of the two knights v. pawn ending comes under the heading "Troitsky line".

I am in total agreement with this paragraph except for the last sentence.

The Troitsky line is exactly as you say a set of squares and is, of itself, associated neither with blockaded pawns nor zones.

It is, however, associated with both by various distinct rules.

Of these rules - the most generally recognised - is commonly referred to as "the Troitsky line rule". It appears in many places in different forms and usually refers to blockaded pawns (though in Fine 1941 it simply says "White wins if the pawn cannot cross the line"). The most accurate statement I know of is that quoted by n95311 from Müller and Lamprecht, which has only 36 positions which break the rule (assuming "blockading" is intended to mean standing directly in front of). Some statements of the rule are wildly inaccurate.

The essential phrase in the M&L version is, "no matter where the kings are". Clearly the Black king can be attacking the blockading knight and the White king can be not defending the blockading knight, so one is led to assume that the the phrase "securely blockaded" means the remaining knight is defending the blockading knight. At any rate I understand that was the meaning intended by the authors.

M&L's version of the rule then doesn't include #1 because the knights are not mutually guarding. But it also doesn't in include all the similar drawn positions with the kings on different squares. It is therefore correct in relation to these whereas all the other versions of the rule I have seen are not.

A second and totally different rule, not known as "the Troitsky line rule" is the one concerning zones I posted earlier. This rule associates the Troitsky line with Black king win/draw zones for (not necessarily blockaded) pawns. This rule was reported by Rabinovich, though he doesn't refer to zones but uses a rather tortuous circumlocution to describe the same thing.  He says, "Troitsky discovered that ...", rather than "Troitsky stated that ..." , so you have to guess from that if Troitsky actually formulated it into a rule. At any rate it is completely accurate as opposed to "the Troitsky line rule".

I stated a third (and different again) rule in my last reply to n95311. I haven't seen this explicitly stated anywhere, but I think Troitsky proved it. This one is also completely accurate.

Hope we're now in agreement on that.