Is this endgame won for white?

Question

With the missing g and e pawns still on the board and a roughly symmetric position, I made a mistake that led to 17 moves of what I think of as quite difficult defense. My opponent walked into the following fork and then resigned.

 [Title "Game's Final Position"] 
 [SetUp "1"]
 [FEN "8/5pp1/7p/1n1R3k/4PP1P/8/8/r3BK2 w - - 6 1"]

I know that I definitely would not have resigned in that position.

Is the above position won for white?

Conceivably, black could play Kg6 and try to put together something like a fortress, or even play g5 immediately after Rxb5. However, I believe the correct continuation would be

1 ... Kg4 2. Rxb5 Kxf4 3. Rf5+ Kxe4 4. Rxf7 g5 5. h5, resulting in the position

 [Title "End of My Main Line"] 
 [SetUp "1"]
 [FEN "8/5R2/7p/6pP/4k3/8/8/r3BK2 w - - 6 1"]

with black to move.

From tablebases (e.g. this site), I know that if I instead played 5. hxg5 and black responded with hxg5 then the position would be a draw. In fact, even if all three remaining pawns disappeared and it became my move, the position would still be a draw. Thus, white's strategy at the end of my main line presumably involves threatening/trying to win the pawn on h6, possibly after exchanging rooks.


(I don't know whether or not the "rook endgame" tag is appropriate, since there's also a bishop on the board.)

Answer 1

According to my analysis, it seems like a clear win. Black can't force the exchange of the remaining white pawn, White has the Bishop of the appropriate color, and Black can't prevent White to improve the position of its pieces.

An important tactical finesse:

1..Ke5 2.Kg2 Rxe1 3. Re7+, winning the Rook.

Another important idea:

1..g4 2. Rf6 g3 3. Rg6! (3. Rxh6 g2+ 4. Kxg2 Rxf1 5. Rf6 =) 3..Ra5 4. Rg4+

and the g pawn falls.

(Note that 2. Kg2 is also possible, with the same idea as above.)

---

It should be a matter of time for Black to lose its remaining pawns, since the black King is cut off from them. I can't offer any concrete lines, so perhaps I'm wrong here. But what I know is that

W: Kf1, Be1, Rh6, h5
B: Ra1, Kf5

is a win in the tablebases. It does not seem to be possible to prevent White to play Rh8 and h5-h6, after which the pawn is immune. As soon as it gets attacked by some Rh2, White has Rg8+, repeating the same tactical theme as above, or Bd2, protecting the pawn directly.

I've played some games with an extra Black pawn, and I have yet to save one game against Stockfish. One annoying bug of having a pawn on g4 is that black King can't move on g4.

Answer 2

Stockfish gives your "end of my mainline position" as winning for white(+2.62). I am attaching the whole line below. It does seem that while the rook protects the bishop, the white king can quickly head over to the black pawn chain.

Score: 2.62
Best: 1. ... Kg4   Ponder: 2. Rxb5
Time: 31.4s   Depth: 16
Server: srv01   CPU #1: 24%
Engine: Stockfish 1.8 64bit

1. ... Kg4 2. Rxb5 Kxf4 3. Rf5+ Kxe4 4. Rxf7 g5 5. h5 Ra6 6. Kg2 Rc6 7. Kg3 Kd5 8. Kg4 Rc4+ 9. Kf3 Rc1 10. Re7 Rb1 11. Bf2 Rb3+ 12. Be3 Rc3 13. Kg4 Ra3

Answer 3

I tried to put up Houdini 6.02 and a tablebase win is reached , albeit it is a mate in 48 !!! According to seven piece tablebases, White is even forced to give up the pawn toward the end but obtain a winning Philidor position! Yes , White wins , but has a lot of work to do. Position WHITE Rook in h6 Bishop in g3, pawn in h6 , King in g2 BLACK Rook in d1 , pawn in g5 King in f5. BLACK to move : White mates in 48 moves.