Today I came across a very nice study in the Chess.com Tactics Trainer. It took me a while to find the solution, but I managed to solve it. To find it in the TT DB, it has problem ID 0044848.

very good...!! this is called forced mate..

cool!

great puzzle!!

That's a good deal beyond me. It's all very clear and logical once you've seen the answer, but to be honest I didn't get close.

Phobetor, may I ask, what was your thought process to be able to find the Bg7 move?  You could have given me a week and I wouldn't have found it.  Great puzzle.

It is an interesting puzzle

Knightsight, some considerations for solving this puzzle:

(1) White will at least draw anyway (a 'small' advantage) but most endgames of R+B vs. R are theoretically drawn. So white has to win more than the pawn.

(2) Black's pieces are badly coordinated, and in particular the king. However, if white moves in to checkmate black with, say, 1. Ba3+ Ke8 2. Ke6, then black always has 2... Rg6+.

(3) 1. Ba3+ Ke8 2. Re7+ Kd8 doesn't lead to anything.

(4) 'Intuition' says that if black manages to get his rook out (g6, g5, g4, g3, g2, g1) and white does not have an immediate win, then white will not be able to do better than reach an endgame of R+B vs. R. So white probably needs a forcing variation.

Then at some point I just came up with this plan of restricting the black rook before advancing the king. And the variations after 1. Bh6+ Ke8 2. Bg7! then show that black is helpless against 3. Ke6 and 4. Rb8+. The only worry you should then have is 2... c2 3. Ke6 Kd8 when 4. Rb8+ Kc7 5. Rxg8 c1=Q doesn't work. But then eventually you see 5. Be5+! and it turns out white wins :)

That's excellent Phobetor.  Great insight.  Very helpful indeed.  Many thanks.  I better get back to tactics trainer and keep working at it. 

nice..