It is very interesting. I do not know the answer. But I think I would play RxR and take my chances with the N v B.
Practical endgame exercise
Thanks for the detailed response JMurakami, I wish I had gotten back to this sooner.
I think a lot of lower rated player's instinct might be to enter the rook endgame with 1...Nxd2 2.Kxd2 because we're told "all rook endgames are draws." However as you point out, white has some serious ideas to try and win in the rook endgame. If black waits with 2...Ra4?! 3.Rc7! (to cut off the black king). Even without a white king on g6, if the white king can get to e5 when the black king is still on the last rank, then Rd6 will win the pawn and the game. So 3...Kh7 but now 4.Rc6 when advancing the pawn weakens it (the white king would only need to come to e4 after moving his pawns off the 2nd rank to deal with counter threats). After 2...Ra4 I think white should win... although I'm not totally sure. I'd be surprised if in a real game a player near my level could defend that as black.
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For that reason and the ones you gave, I dislike 1...Ra4, keeping the rooks on the board. I think the minor piece endgame offers much easier drawing chances with 1...Rxc1 2.Bxc1 and now what is white's winning idea? If black tries to set up a fortress with pawn on e6, knight on d7, king f7 then white pawns on e5, f4, g4, h5 and the white king pushes in from the queenside Kc2-d2 and zugzwang pushes white's pieces backwards. Importantly the black knight wont be able to infiltrate to white's side to threaten pawns.
So what is black's idea in this ending? Black needs space for his knight so that when the white king comes far up the board, the black knight can escape via the queenside and infiltrate to white's position, and attack the pawns. First of all black obviously wants those pawns off dark squares.
2...h5 3.Bb2 g6 4.Ke2 Kf7 5.f3 Nc5 6.Bd4 Nd7 7.Kd3 e5 8.Bc3 Ke6
In that position after 8...Ke6 the idea is the black king anchors squares like d6 and d7 which the knight will use to go to b6 or b7, threatening to infiltrate to white's side, when white tries the winning method outlined above. I think this is a pretty easy fortress.
So white might try 7.f4 to cramp black but 7...Nb8! this is the correct side. 8.Kd3 Ke7 9.g3 Nc6
And black will shuffle the knight between the squares c6 b8 and a6, and the king between squares e7 d7 and e8 for an easy draw. The black knight will not try to infiltrate until the white king has at least crossed the mid line.
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Maybe it's too much to say which is easier, the rook endgame or minor piece ending. But if done correctly, I think the minor piece ending offers much less winning chances for white. Only able to hit half the squares with the bishop, the weakness of black's pawns are reduced, and white's winning options are limited.
Black is pondering 1. ... Rc4xRc1+ or 1. ... Ne4xBd2 or 1. ... Rc4-a4
shall we swap "e" for "g" and white has a passed pawn:
1. ... Ra4 2. Rc8+ Kh7 (JM) 3. Bc1 Ra2 4. f3 Nd6 5. Rc6 Nf5 6. Bd2
is probably easier to grasp.
1. ... Nxd2 2. Kxd2 Ra4 3. Rc7 and white can play King to f3. Not sure if black can hold with e5,e4
It really looks like 1. ... RxRc1+ by process of elimination.
1. ... Rxc1+ 2. Kxc1 h5 3.Bb2 g6 4.Ke2 Kf7 5.f3 Nc5
After that I do not know but not kick the knight to a white square.
They say all rook endings are drawn. I'd take R v R. Since the pawns are all on the same side of the board, the knight might be stronger than the bishop, but I'd still rather play R v R.
In a tournament game, the clock situation would be important as well. If I was up time, I would keep the position as complex as possible. If not, R v R
#6 : 1. ... Nxd2 2. Kxd2 Ra4 3. Rc7 is what OP thought about the rook endgame.
Do you have a line in mind please?
Minor piece ending a pawn down and a weak one as well too. Right. Don't trust engines in positional games and endings.
White sets his Bishop in a3, cutting the Knight's retreat and coordination with his pawns and King. Pawns in f3 and e4, King to f4, threatening Kg5 and a rupture in f5, picking up that nice little fellow engines suggests to move to h5. That line when White plays f2–f4, wow. He himself takes in charge the security of Black's kingside pawns by blocking the roads to both his King and Bishop.
The treatment is simple: Getting a passed pawn, diverting the defenders and going after the weaklings on the right hand.
For example: 1...Rxc1+?! 2.Bxc1 h5?! 3.Ba3 Kf7 4.f3 Nf6 5.e4 e5 (necessary to deal with the threat mentioned above) 6.Kf2 Nd7 7.Bb4 Ke6 8.g3 Nf6 9.Ke3 g6 10.Bc3 Nd7 11.f4 (passed pawn) exf4+ 12.gxf4 Nc5 (must keep the White King tied to e4 as long as possible) 13.h3 Nb7 14.Bb4
I can't guarantee White wins. What I can guarantee is that Black will have to play with precision for some dozens of moves, just to not to lose.
I rather take my chances with Rooks
The engine doesn't suggest h5 (or the minor piece ending) this is just my idea.
I also looked at setups with e5 for black, but I think they fail. In your line after 7...Ke6 I think white can win with 8.Kg3 planning Kh4 with g3-f4 advance to get a passer much like you suggest.
I think h5-g6-e6 is the best formation for the pawns... but your Ba3 is very annoying because my king wants those squares like you say
I'll have to look at it some more... but my initial impression is you're right that there is a big demand for accuracy from black.
From an evaluation point of view, would agree with JMurakami. Would try to hold the rook endgame 1...Ra4. (I also don;t see why not kf7 then try to use rook to cut off white king, but that's not the main point). From an evaluation point of view, without going into details, would be expecting the pawn up, better pawn structure, BvN endgame to be a win.
Is the OP getting this endgame excercise from a book, or calculating with software, or what?
I made this up based on a game and some analysis I saw of a B vs N endgame. Rook endgame is the obvious answer, but the minor piece endgame isn't as easy as I thought. And no, of course I wouldn't just dump engine analysis. The engine doesn't even recommend my line, It tries to play some losing moves early and don't realize they're bad until much much later.
Here black is a pawn down and has a structural defect on e6.
Black also has the choice between 3 different endgames:
R+N vs R+B
R vs R
N vs B
What's the best endgame to fight for your draw?
You could also think of this the other way because sometimes you'll be the one with an extra pawn. If white got to choose between those 3, which endgame lets white fight best for the win?
After testing out different defenses, I think I'm happy with my answer. I thought others might find this useful or interesting.