ConstantImprovement: PART I:
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.
b223. 24. ... Kf7 25. Rf1+ Ke7 26. Qh7+
(26. Qf8+ Kd7 27. Rf7+ and the Black King escapes via c6) ... Ke8 (26. ... Kd8 Rf8#) 27. Qc7: and now it is like at b222, 27. ... Nd7 28. Rd7: Bd7: 29. Qb8:+ Ke7 30. Qb7:, with the same winning ending.