Can this exchange-down endgame be explained succinctly? I am interested in the following exchange-down endgame (assume White to move):

[fen "6k1/5ppp/8/8/1r6/5KP1/5P1P/5B2 w - - 0 1"]


In particular, I am wondering whether this is objectively drawn (my hunch) or whether Black has a winning advantage. But really I'm after more than just the assessment of the position.


  My real interest is in knowing whether a drawing method for White or a winning method for Black can be explained in a relatively simple way. (If it can, I'm hoping for a demonstration of that fact as an answer.)


I'll say a little more about what I'm after. (First, what I'm not after: a synopsis of your engine's analysis. I have engines too. And the fact that, e.g. Stockfish gives this position -3.00 or so just doesn't signify all that much.) Consider the endgame where Black has no men left, while White has a rook pawn and the wrong-colored bishop. The drawing method there (assuming the black king has access to the queening square) is so simple that it can be quickly explained to a novice, and she'd be able to hold against any world champion.

The endgame I'm asking about here is more complex, of course, and any reasonably complete explanation of how to handle it will be correspondingly more complex as well. But I'm still hopeful that a fairly concise explanation of how to handle it (either how to win for Black or how to draw for White) can be given; and like the rook-pawn-and-wrong-bishop ending, I'm hoping for the sort of explanation that a relatively weak player could use as a reliable recipe for correct play.



No doubt, I'm setting a high bar for an answer here, but I think a good answer along the lines I'm asking for would be very valuable. And even if this particular position ultimately can't be explained well in this way, in more general terms, I think a large body of solid questions about common sorts of positions like this one, and especially answers offering clear, short verbal explanations of these positions, would be well suited to this site, and could also make it a useful reference, offering a sort of free, collective middlegame and endgame manual.
This endgame is completely winning for black with best play.  While there are some cases where the weaker side can build a fortress, those only occur with 1 or 2 pawns per side, not with 3 undoubled pawns.  

For example, here is the most common fortress (with 2 pawns each):

[FEN "6k1/1R6/4K1p1/7p/8/2b3P1/7P/8 w - - 0 1"]


White cannot break down black's defense.  If the white king tries to come to g5, black prevents this with ...Bf6! when the white king appears on f4.

Dvoretsky's endgame manual explains succinctly:


  With three pawns on each side a fortress, as a rule, cannot be build.  Salvation is possible only in exceptional cases: when the pawn structure of the stronger side has flaws.


As an extreme case, look at the most "advantageous case for the weaker side, with the pawns in contact (the variation is from an actual game while the main line is black's best try):

[FEN "6k1/3R4/6p1/5p1p/5P1P/3K2P1/1b6/8 w - - 0 1"]

1. Kc4 Kf8 2. Kd5 Kg8 3. Ke6 Bc3 4. Rd3 $1 Bb2 5. g4 $3 $18 fxg4 (5... hxg4 6. h5 Kg7 7. hxg6 Kxg6 8. Rd5 Bc1 9. Rxf5 Bxf4 10. Rxf4 Kg5 11. Ke5 g3 12. Ke4 g2 13. Rf8 Kh4 14. Rg8) 6. f5 gxf5 7. Kxf5 Kf7 8. Kg5 Be5 9. Kxh5 g3 10. Rd2 Kf6 11. Kg4 Kg6 12. Re2 Bb8 13. h5+ Kh6 14. Re6+ Kh7 15. Rg6 Bc7 16. Kf5 Bb8 17. Rb6 $1


Now that we've established that the stronger side can win, even with the pawns in contact, let's look at the case in the question:

Black's winning plan is quite simple, bring the king to e1 to attack the pawn on f2 with both the rook and the king (forcing its advance to f4), and then black will bring the king back to e3, bring all 3 pawns to the 3rd rank, and finally push g6-g5, forcing weaknesses.

Here is a sample line.  It is just a single example, there are many other options for both colors, but hopefully it illustrates the winning plan for the stronger side.

[FEN "6k1/5ppp/8/8/1r6/5KP1/5P1P/5B2 b - - 0 1"]
[PlyCount "41"]

1... Kf8 2. Ke3 Ke7 3. Bd3 g6 4. Kf3 Kd6 5. Ke3 Ke5 6. Kf3 Kd4 7. Be2 Rb3+ 8.
Kg2 Kc3 9. Bf3 Kd2 10. Bc6 (10. Bd5 Rb2 11. Bxf7 Ke1 12. Bg8 Rxf2+ 13. Kg1 h6
14. Bd5 Ke2 15. Be6 Rf6 16. Bc4+ Ke3 17. Bb5 h5 18. Kg2 Rf2+ 19. Kg1 Rc2 20.
Ba4 Rc4 21. Bb3 Rd4 22. h4 Rd2 23. Bf7 Kf3 24. Bxg6 Kg4 25. Kf1 Rd6 26. Be8 Rd8
27. Bb5 Kxg3 28. Ke2 Kxh4) 10... Ke1 11. f4 Rb2+ 12. Kg1 Ke2 13. Kg2 Ke3+ 14.
Kg1 f6 15. h4 h6 16. Bd5 (16. Ba8 g5 17. fxg5 hxg5 18. hxg5 Rb8 19. Bc6 Rb6 20.
Ba8 f5 21. g4 f4) 16... g5 17. fxg5 hxg5 18. Bc6 (18. hxg5 Rb5) 18... gxh4 19.
gxh4 f5 20. h5 Rb6 21. Be8 f4 0-1


To finish this answer, consider the position if white has instead a dark squared bishop.  In this case, black's plan is very similar - attack the f2 pawn.  Black brings the king to e2, the rook to f3, and then advances the pawns (g7-g5, f7-f5-f4 - potentially including h7-h6 if needed) to break through.

Hopefully this gives you all of the plans needed to win as the stronger side.  The defensive chances with fewer than 3 pawns each are also worth examination, but they're probably worth a question on their own.