What is the most difficult winnable endgame to learn?

This is a position that is mate in 108 (I think) with the 50 move rule in effect (only 85 without). You could try that one to start; after ...Ka4 White has only one move to win with the 50 move rule in effect. When you find out why, let me know. (The initial move in the 549 mover is more obvious.)

I don't trust the knight endgame tablebases. 1 king + 1 knight is just not enough to force the king to the edge, let alone toward a corner. It's not like the other endgames where there's a finite number of geometric piece configurations and eventually the defensive ones get used up. How does the king and knight actually trap the king while the other knight is stuck guarding the pawn?

https://www.chessgames.com/perl/chessgame?gid=1295765

Here's a game between two top players back in 2000 that ended in the 2 knight vs pawn endgame. Wonder if the moves after kc3 were "perfect play" according to tablebases, since the pawn was past the Troitsky lime, but white still managed to win.

What's missing in this thread are the players. Ask them and you will find that most players have zero interest to play endgames that last forever and they don't want to play games that last forever. Why not set a ceiling of like 150 moves for the sum of them, the complete game? And make that end automatic such that the players can't continue in "trolling mode".

The tournament directors will be happy as they need not screw the players into blitz increments causing unfair losses by stumbling players.

And on the artistic side we lose nothing since these game limits need not apply to compositions - just as it is now!

Btw, limits like 600 moves do not come from some objective chess criterion but purely from the current state of tablebase knowledge. In 10 years time they completed the 8-piece section and there is a new maximum. Why do limits for game players have to change when theory grows in places which players can never learn anyway?

You prejudice right there and you're ignorant and insolent. Don't claim someone else can't do what you believe you can't do.

Most arbiters do the same. They are entitled to judge the games between the players who know more about the game then they do. It is the same on every field. It's not surprising the society is in decay. Politicians should go to war instead of millions of people and there will be no wars.

The other day they fined a girl because of the sport shoes she wore?

I would say place a total sum time limit on the game in addition to whatever time control increment is being used. When the game is taking forever, just say the game has to end in a half hour or something, regardless of the clock time, no decisive result in that time = Draw. A 150 move limit would have interfered in some of the greatest long chess games ever played. World Championship 2021 Game 6 was 136 moves and could have easily lasted longer (without even tripping the 50 no-pawn-no-capture-rule).

it is 2 knights vs pawn

The 150 number is only an example. If 200 is better, then it is 200. Actually I think the limit could be made dependent on the type of tournament you're playing in. Need not be a fixed number and the number will probably be larger for a WC-match than in some preliminary round.

Time controls are easier to manipulate by the players than move-number controls. For instance one player might let the 30-minute extension run out if that gets him the draw. Increments are in itself a dubious system as they change the character of the game while it's running. You came to play a slow game and you end up blitzing to keep up with the tiny increments. Of course, it's decent when the increments are pretty big as well but that might not achieve the goal. In the old days you got fixed hourly increments for 20 moves until adjournment which was most fair. Today adjournments are impossible and everyone is scrambling for some solution.

b+n v 2p+r

They made three half point blunders between them in the first nine moves of the endgame. From the grandmaster games I've looked at where the ending occurs, that's par for the course.

By the way there's nothing about the Troitzky line that precludes winning positions for the knights with the pawn beyond (nor in fact drawn or winning positions for the pawn with the pawn on or behind) if that's what you were implying.

Wrong, I'm afraid.

I think I suggested it once before; invest in a copy of Troitzky's COLLECTION OF CHESS STUDIES ISBN 0-923891-10-2. There's a 60 page supplement with a comprehensive analysis of the White wins in this endgame (written decades before any tablebases). Expect to spend some time and don't expect any help with the Black wins or, from a theoretical perspective, positions like the one I posted.

Much work has gone into checking the veracity of the tablebases. I think you can trust them.

Edit: Up to 5 men at any rate. The fact that the Syzygy site no longer shows DTM values for 6 men and above suggests checking has been less comprehensive.

You only have to try defending, for example, the position I posted against Syzygy here to see the error in your analysis. You will find yourself mated within the 50 move rule in a corner from that position. (Why are you assuming it 's a corner from all positions? There are forced mates anywhere around the perimeter except on the two central files - though, if I remember rightly, only from illegal positions on the pawn side's bishop and knight's squares. Admittedly many of them are not very generally useful, but some are essential.) If you try it against Nalimov you may or may not find yourself mated within the 50 move rule (based on which of Nalimov's moves are played, not your understanding of the endgame), but I'd guess you would, based on what you've said so far.

If a player was able to run his time like that, I doubt the 50 move rule could have even kicked in yet. And if they moved so fast that they both still had a half hour left on their clocks after 100+ moves, they'll likely end up hitting whatever arbitrary number is set anyway. I feel the increment should actually change as the time decreases. Maybe only a 1 sec increment over 5 mins and then a 2-3 second one under 5 minutes. I read something called a "Bronstein delay" which is not an increment. It doesn't add time, but the delay is based on how long you spent on each move. So you can never gain back more time than you started with via increment and pointless shuffling, but you get the same delay to avoid unfair flagging.

Limits on the required n in an n move rule don't actually come from current tablebases. They don't tell you. Nor would any currently under construction, so far as I know.

Note that the positions with the greatest distance to mate under basic rules are not necessarily the positions that would require the greatest allowance under an n move rule, nor even are the positions with the greatest DTZ metric.

I wasn't referring to rules governing the no-progress phases without captures or pawn moves but the complete duration of any endgame - which could as well be extended to the whole game which makes the numbers only slightly larger. Somebody suggested 600 earlier in the thread to cover the longest 7-piece endgame, .... and will suggest another number when tablebase research reveals more. I only know for sure it will never top 10,000 under the 50-move rule.

Run like what? The scenario with 20 moves per hour did exist, so it's not theoretical. And why wouldn't the 50-move kick in? Many endgames are "easy" for one side and "hard" for the other. The side with the upper hand can often accumulate time when the increments are better than blitz speed. Bronstein's system prevents that but is harsh on the player having saved time during the game. Perhaps one could distribute the time saved before the "Bronstein phase" over the remaining moves such that the player with more saved time also gets a bit more time for every move while on "Bronstein" (until it's spent of course).

K + 2 Knights + Bishop vs King + 1 Rook is not a win. wikipedia says it is a draw because the defender can simply attack the bishop with the rook and threaten to sac the rook, as 2 knights vs king is a draw

Yes that's why 2 bishops + knight win, because whatever the rook takes the other 2 can win. Not the case with 2 knights + bishop, correct.

Never believe Wikipaedia. The tablebase says over 20% of positions are won by White.

Whether K + 2 Knights + Bishop vs King + 1 Rook is a win depends on where the pieces are.