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I encountered the following position with White to move.
[FEN "6R1/8/3k4/pK6/Pb6/8/8/8 w - - 45 83"]
I see no way for White to win, but am not completely sure. Is it known to be a draw? More generally, is the result for all such positions with head-to-head pawns and rook against bishop known?
This endgame is indeed a draw. You can check positions with 7 or fewer pieces in the Sygyzy tablebases to see if they are a win, a loss, or a draw. If you want the depth to mate for 6 pieces or fewer, use the Shredder database
Given the context, I suppose you would like to know why this is a draw. If White was trying to win, the best option is to take Black’s pawn. Black’s bishop is stuck defending it. If the pawn falls, Black falls with it. White can and will win the pawn.
[FEN "6R1/8/3k4/pK6/Pb6/8/8/8 w - - 0 1"]
1. Ra8 Kd5 2. Rxa5 Bxa5 3. Kxa5 Kc5
The clutch is doing soh leaves White with a major problem. The pawn endgame is a draw! Due to having a rook pawn, White will either be stalemated by the Black king, or Black will get in front of the pawn and be stalemated by White.
[Title "White is stalemated"]
[FEN "8/8/8/K1k5/P7/8/8/8 w - - 0 1"]
1. Ka6 Kc6 2. a5 Kc7 3. Ka7 Kc8 4. a6 Kc7 5. Ka8 Kc8 6. a7 Kc7
[Title "Black is stalemated"]
[FEN "8/8/8/K1k5/P7/8/8/8 w - - 0 1"]
1. Ka6 Kc6 2. Ka5 Kb7 3. Kb5 Ka7 4. a5 Kb7 5. a6+ Ka7 6. Ka5 Kb8 7. Kb6 Ka8 8. a7
If White does not target g the Black pawn, BBlack will merely wait it out. They will just shuffle their bishop around to keep protecting their pawn. The Black king will move around such that the pawn endgame will remain drawn. Eventually, there will be a draw via the 50-move rule or threefold repetition.
For all situations with head-to-head pawns, it seems that most will be won. Moving your position one file over move creates gives a win for White. Therefore, it seems to me that the only drawn positions are ones with rook pawns and a close enough Black king. In other words, there are very few of them.
[FEN "7R/8/4k3/1pK5/1Pb5/8/8/8 w - - 0 1"]
1. Rb8 Bb3 2. Rxb5 Kd7 3. Rb7+ Kc8 4. Re7 Bd1 5. b5 Bf3 6. Rf7 Be4 7. Kb6 Bg6 8. Re7 Kd8 9. Re5 Bc2 10. Kc6 Bb3 11. b6 Ba4+ 12. Kd6 Kc8 13. Re7 Kb8 14. b7 Be8 15. Kc5 Ka7 16. Rxe8 Kxb7 17. Re7+ Kc8 18. Kc6 Kb8 19. Rf7 Ka8 20. Kb6 Kb8 21. Rf8#
This particular position is a draw; since it's a 6-man endgame, you can use e.g. Shredder's database.
However, not all positions like this (with blocked rook pawns) are drawn. A famous one is Timman vs. Velimirovic, Rio de Janeiro, 1979. That game was adjourned three(!) times, giving both players plenty of opportunity to analyze it. In the end, Timman managed to win it, just before the 50-move rule kicked in. The main idea is to cut off the black king on the kingside (by mating threats and/or attacking the bishop), then sacrifice the rook for the bishop and the pawn and use the king to block the black king from reaching the promotion square.
Again, that is not possible from your starting position, since the Black king is already close to a8, but with the Black king on f4, it is a win for White:
[FEN "6R1/8/8/pK6/Pb3k2/8/8/8 w - - 45 83"]
1. Ra8 Ke5 2. Rxa5 Bxa5 3. Kxa5 Kd6 4. Kb6 Kd7 5. Kb7
The same procedure (sacrificing the rook for the bishop and pawn) always wins if the pawns are not rook pawns. It's relatively easy for White to reach one of the key squares of the last pawn after the exchange, except if you time it so badly that the Black king can take opposition. But it's rather simple to drive the Black king far away enough.
I once drew a game by heading to this type of position. A great benefit of knowing endings is knowing when to head into the ending and what positions to aim for. I am black below in a tense game with blunders on both sides.
[Result "1/2-1/2"]
[Termination "Game drawn by agreement"]
[TimeControl "1/259200"]
[FEN ""]
1.c4 Nf6 2.d4 g6 3.Nc3 Bg7 4.e4 d6 5.Nf3 O-O 6.Be2 e5 7.d5 a5 8.h3 Na6
9.Bg5 h6 10.Be3 Nh5 11.Qd2 Nf4 12.Bxf4 exf4 13.Qxf4 Nc5 14.O-O-O Re8
15.Nd4 Bf6 16.h4 g5 17.Qe3 gxh4 18.g3 Qe7 19.gxh4 Bg7 20.Rdg1 Kh8
21.Bh5 Qe5 22.Bxf7 Re7 23.Nf3 Qf6 24.Bg6 a4 25.e5 Qf8 26.e6 a3
27.b3 Qf6 28.Kc2 c6 29.Nd2 cxd5 30.cxd5 Nxe6 31.Nde4 Nd4+ 32.Kb1 Bf5
33.Bxf5 Nxf5 34.Nxf6 Nxe3 35.Rg6 Rf8 36.fxe3 Rxf6 37.Rxf6 Bxf6
38.Nb5 Rxe3 39.Nxd6 Rd3 40.Kc2 Rxd5 41.Nxb7 Be7 42.Re1 Bxh4 43.Re6 Kg7
44.b4 Bg5 45.Kb3 Bc1 46.Re1 Bb2 47.Na5 Rd3+ 48.Ka4 Rd4 49.Nc6 Rc4
50.Ne5 Rd4 51.Rh1 Bc3 52.Nc6 Rc4 53.Kb5 Rxc6 54.Kxc6 Bxb4 55.Kd5 Kg6
56.Rh3 Kg5 57.Ke5 h5 58.Rg3+ Kh4 59.Rb3 Bf8 60.Kf5 Be7 61.Rb7 Bd6
62.Rh7 Bc5 63.Kg6 Kg4 64.Rxh5 Bd4 65.Ra5 Bb2 66.Rf5 Bc1 67.Kf6 Bb2+
68.Ke6 Bc3 69.Rf7 Kg5 70.Kd5 Bb2 71.Kc4 Kg6 72.Rf1 Kg5 73.Kb4 Kg6
74.Rf3 Kg5 75.Rxa3 Bxa3+ 76.Kxa3 Kf5 77.Kb4 Ke6 78.Kb5 Kd7 79.a4 Kc7 1/2-1/2
