So, let us look at the basic motifs of the position and see if our final solution will be somehow made of all we noticed:

1. The Bishop check on b5
Since Kf8 is no option because of Rd8+ Qd8: Qd8:#, only Nd7 and Bd7 remain, both blocking the Queen's guarding of f7.

2. The weak f7-spot
Rf1 is directly staring there, Be2 can be there in one move. A sacrifice at f7 might be possible. Perhaps Bb5:+ before it to block the Queen's guard (see above).

3. The mate threat at d8
The Queen and the rook at d1 are having a close look at d8, forcing the Black Queen to guard this square.

4. A possible square for the Queen at f6 after Black's g6 This would reinforce the d8-threat and allow further attacks.

Now let's bring all that into motion:

I. 20. Bb5+

1. 20. ... Nb5 21. Rf7: (Moves like 21. Qf4 probably are too slow after 21. ... Rf8)

a. 21. ... g5 22. Qf2 with Qf6 and a mate on e7 (22. Qh5 is the same)

b. 21. ... Kf7:

b1. 22. Rd1+ Ke8 (22. ... Nf6? 23. Rf6:+ f6: 24. Qf6:# or 23. ... Ke7 24. Rf4#)

b11. 23. Qf4 does not seem to be strong enough, since Black (being a rook and a piece up) can simply play 23. ... Qb6 and allow 24. Qf7+ Kd8 25. Qg8:+ Kc7, where Black should not be worse with pressure on the a-file and a piece up.

b12. 23. Qh5+, and either 23. ... g6 24. Qf3 Qb6 or even perhaps even better 23. ... Kd8 24. Qf7 Rd8, with Black winning

And now a little help for analyzing:
We now saw during our analysis that there are sufficient defensive maneuvers (Qb6 and others) against a threat of the Queen and d1-Rook on the f-file. So we will not analyse lines like 22. Qf4+, 22. Qh5+ or 22. Qf2+, because they will probably transpose, having the same attacking ideas.

Let us go further back.

We will ignore 20. ... Bb5, too, since it is not clear how this should help White's attack. We will now try to bring the White Bishop to attacking the f7-square (See motif 2).

II. 20. Rf7:

1. 20. ... Qf7: 21. Rd8#

The Black Queen needs to guard the mate on e7, which she can only do from c7 (see the d1-Rook against Qd8 and Qd6). So every other move with the Queen results in mate. Help might come from

2. 20. ... g5, intercepting the diagonal, but there follows 21. Rc7: h4: 22. Bh5+ Rg6 23. Bg6:+ Ke8 24. Rd8#

So Black has to take the Rook:

3. 20. ... Kf7: 21. Bh5+ (Other lines like 21. Qf4+ Ke8 are too much like the "20. Bb5:+"-lines to be successful.)

a. 21. 21. ... Kf8 22. Rf1#

b. 21. ... g6

b1. 22. Rf1+ Ke8 is too slow

b2. 22. Qf6+ Ke8 23. Bg6:+ (Other moves are too slow, too) 23. ... Rg6:

b21. 24. Qg6:+?

b211. 24. ... Kf8 25. Rf1+ Ke7 26. Rf7+ Kd8 (Ke8) 27. Qg8#

b212. 24. ... Ke7 25. Qg7+ Ke8 26. Qc7:, winning the Queen

b213. 24. ... Qf7 25. Rd8+ Ke7! (25. ... Kd8: 26. Qf7:, with a slight advantage for White), and Black has a winning position

b22. 24. Qh8+

b221. 24. ... Rg8 25. Qg8:+ Ke7 26. Qg7+, winning the Queen on c7

b222. 24. ... Ke7 25. Qh7+ Ke8 (Kf8) 26. Qc7:, threatening mate on d8 and the Rook on b8; 26. ... Nd7 27. Rd7: Bd7: 28. Qb8:+ Ke7 29. Qb7: with a winning ending Q+6P against R+B+3P, which White will win after 29. ... Rg2: 30. b3 Rh2: 31. a4 a4: 32. a4:, since the Bishop cannot take the pawn and the Queen should stop the h-pawn in time. After moves like 29. ... Ke8 or similar to prevent a4, White plays 30. g3 and now has two possibilities for a passed pawn.

So let us conclude:

Of our motifs we saw at the beginning, one had to be discarded and the other three played crucial roles in our main line. We can add "the unguarded Queen" and "the decoying check" (even though Rd8+ did not work because of Ke7). Motif 2 guided us to abandon the "20. Bb5:+"-lines for 21. Bh5+, which proved to be the right way.

I hope to have helped especially beginners to understand the logic underneath most chess combinations.

At the end our main line:

I did, but when I couldn't find a mate in the last line (only the advantage of queen and pawns against rook + bishop), I was concerned that I had missed something obvious, as all other lines ended quickly.

I also spent nearly an hour on the position (but some of that time was in writing my entry).

20.Rxf7. Any defence besides Kxf7 seem to fail due to mate on e7 and d8 and an attacked Queen on c7.

21.Bh5 g6 22.Qf6+ Ke8. All forced, leading to first key move 23.Bxg6+ Rxg6 and finally, and here I failed (with 24.Qxg6+ Qf7 25.Rd8+ Kxd8 26.Qxf7 with a very complex, but probably drawing position), 24.Qh8+. There just aren't that much variations, and I just plainly missed the back rank check and mate (or enormous material gain).

I easily noticed the mate threat (Qd8#) that would occur if only I could deflect the black queen. The only possibility along those lines is 20. Rxf7!, but of course, black (unless he's a fool) will recapture with 20...Kxd7, exposing his king rather badly. This is the line we must pursue.

The attack continues with 21. Bxh5+ (I considered 21. Rf1+, but the threats dry up after ...Ke8). Black must play 21...g6 (if 21...Kf8 22. Rf1#), but this opens e6 for 22. Qe6+, and again black is forced: 22. Ke8. Then 23. Bxg6 forces 23...Rxg6.

Here is where we have a decision: First, I looked at 24. Qxg6+ Qf7 25. Rd8+ Ke7, and white is stalled. I can't afford to let black's queen come to the defense. The other option, 24. Qh8+! was more promising (although my board vision is getting blurry now).

26...Kd7 27.Rd1+ Kc6 [27...Nd3 28.Rxd3+ 29.Kc6] 28.Rd6+ Qxd6 29.Qxd6#>

27. Rd1+ is an excellent idea, but you should have analyzed the 27. ..Nd3+ line a bit deeper - the sacrifice gives the black king a new escape square on c5. But now White has 28. Rxd3+ Kc6 as you mentioned (I don't think you actually mean 29. Kc6 though), followed by:

29. Rd6+ Kc5 30. Qf2+ Kb4 31. c3+ Ka4 32. Qa7+ Qa5 33. b3+ Ka3 34. Qxa5#

No, wait, instead of 32. Qa7+, 32. b3+ followed by either 32. ..Ka5 33. Qa7# or 32. ..Ka3 33. Qb2# is faster. 32. Qc2+ Ka4 33. b4# also works.

Here's one, me, who would not have had the nerve or the board vision to play 19. Nb5 as the "set-up" sacrifice for this entire line.