1 Rh5 Bd7+ 2 Kg2 Kf4
The only way for Black to make things evolve;
The purpose of this move is to prepare a Perpetual Check from the 8th rank;
3… Bc6+ 4 Kh3 Bd7+ 5 Kg2 Be6!?
The meaning of this move is to prevent the vertical Perpetual Check by guarding g8; if 5… Bg3? 6 Rf8+ Kg4/Kg5 7 Rg8+ Kf4 8 Tf8+ Ke5 9 Rf7! Bg4 10 Rg7 Draw
Here Black has another clever defence which will be examined below: 5… Bb5!?
Of course not 6 Rf8+? Kg4! 7 Re8 Bd5+ etc.
6… Bd5+ 7 Kh3 Bg3 8 Rf8+ Kg5
If 8… Ke4 9 Re8+ (for example) Kf5 10 Rh8 Kg5 11 Rh5+ Kxh5 stalemate
9 Rf5+! Kxf5 model stalemate
Black has another interesting possibility at 5th move, leading to a long variation:
5… Bb5!? 6 Rf8+ Kg5 7 Rg8+ Kh5/Kh6 8 Rh8+ Kg6 9 Rg8+ Kf6
If Black chooses to prevent the Perpetual Check by playing the King on f7, then it follows an interesting sub-variation:
9… Kf7 10 Rc8! (Not: 10 Rb8? Bc6+ 11 Kh3 Bg3! 12 Rb3 Bd7+ 13 Kg2 Bc7! 14 Rh3 Bc6+ etc.) Bg3 (here after 10… Bb4? 11 Rh8 Be7 12 Rh7+ Draw) 11 Rc3! Bd6 (11… Be5?? 12 Rc5!) 12 Kh3 Be7 13 Rc7! Bf1+ 14 Kg4 h3!? 15 Rc3! (Not 15 Rc2?? Bg2! Nor 15 Kg3? Ke6!) h2 16 Rf3+! and 17 Rxf1 controls h1 and Draw.
10 Rf8+ Kg7 11 Rc8! Bb4!?
If 11… Bg3? 12 Rc3 Bd6 (or 12… Bf4 13 Rc5 Bd7 14 Rh5! Bg3 15 Rg5+!) 13 Kh3! Be7 14 Rc7! Draw.
Not 12 Kh3?? Bd7+! Nor 12 Rb8? Bc6+ 13 Kh3 Be7 14 Rb6 Bd7+ and for example 15 Kg2 Bg4 16 Rc6 Bf6 17 Rc4 Be6 18 Rc7+ Kg6 etc.
If 12… Kg6 13 Kh3 Bd6/Ba5 14 Rb7 Draw.
13 Kh3 Be1 14 Kg4!!
This is a typical draw position: White can just play the King between h3 and g4, waiting for a black attack; the main chances for Black is to find a way to play the Bishop on diagonal c8-h3 with the aim to put aside the white King. The Rook plays to prevent this: here it controls d7 and c4 (preventing Bc4 then Be6+), and also a6; we are going to see that playing the Bishop on d3 (threatening Bf5+) leads also to a draw, but first let us see what happens if the other Bishop plays to g3 or f2, or if the black King tries to protect the Pawn:
Not 14… Bg3/Bf2?? 15 Rh7 threats 16 Rxh4 Draw.
15 Kh3 Bf1+ 16 Kg4 Bb5
We get the same position but now with white to play.
17 Kh3! Bg3 18 Rh7 Be1 19 Rc7! Bf2 20 Rh7!
Not 20 Kg4? Be2+ 21 Kh3 Bd3! (threats 22… Bf5+) 22 Rc3 Bf1+! 23 Kg4 Be2+ 24 Kh3 Kg5/Kf5 wins.
20… Kg5 21 Rg7+ Kh5 22 Rh7+
This prevents the black threat 22… Bf1+
22… Kg5 23 Rg7+ Kf6
If 23… Kh5 24 Rh7+, an echoed perpetual check! If 23… Kf5 24 Rf7+!
24 Rh7! Be1
A significant sub-variation here, which explains why the Rook can play on h7 and leave the control of c4 (and also c5 if the Bishop plays on d3 as we will see below):
If 24… Bc4!? 25 Kg4! Be2+ (if 25… Be6+ 26 Kf3! Bg3 27 Rh6+!) 26 Kh3 Bd3!? 27 Rh8!! Kg5 (if 27… Bf5+ 28 Kg2! Bg3 29 Rf8+! echo! If 27… Kg7 28 Rxh4 Bf5+ 29 Rg4+! [checking] Kf6 32 Kg2 Draw) 28 Rg8+ Kh5 29 Rh8+ Kg5 30 Rg8+ (perpetual check) Kf5 31 Rf8+ Draw.
This underlines why f2 is a “bad” square for the black Bishop!
25 Rc7 (Positional Draw) Bd3 26 Rc5!!
In spite of the “!!”, there is minor dual here: 26 Rc1; “minor” because after 26 Rc1 Bg3 27 Rc5 (the only move) Bf2 we come back to the main variation, so the main idea is there.
We get here another typical draw position: Rook controls all the squares which could allow to the Bishop to reach the diagonal c8-h3!
26… Bf2 27 Ra5!!
The Rook can leave the control of c4 because in f2 the Bishop is in weaker position:
27… Bc4? 28 Ra4! Be6+ 29 Kg2 Bg3 30 Ra6! Draw (also a kind of echo with the above sub-variation); this variation explains also why a5 is the only square.
If 27… Bf1+ 28 Kg4 (threats 29 Rf5+) Be2+ 29 Kh3 Bc4 30 Ra4 Bf1+ (if 30… Bb5?? 31 Rf4+! Or 30… Be6+ 31 Kg2 Bg3 32 Ra6) 31 Kg4 Bg3 32 Ra3 Draw.
Again the Rook must control c4; if 28 Ra3? Bf5+ 29 Kg2 Bf4! 30 Ra6+ Ke7 31 Ra7+ Ke6 32 Ra3 (to control h3) Bd6 33 Re3+ Kd5 etc.
28… Bf2 29 Ra5 Bg3 30 Rc5 (echoed Positional Draw) Kg6 31 Kg4!
White is waiting!
31… Be2+ 32 Kh3 Bd3 33 Kg4 Be2+ 34 Kh3 (Positional Draw) Kf6 33 Rc3! Bf1+ 34 Kg4 Draw.
(Also possible at 30th move, although less interesting: 30… Bf1+ 31 Kg4 Be2+ 32 Kh3 Kg6 33 Rc3 Bf1+ 34 Kg4 Bd2+ 35 Kh3 Perpetual Check)
Besides the aesthetic content (mainly model stalemate in 1st variation, successive sophisticated echoed positional draws in the 2nd variation), I do think this study has also a theoretical interest: it shows almost all the existing Draw possibilities and strategies for White with this material in this specific case: Pawn is on h4, protected by a Bishop on diagonal e1-g3, Black King being on files f to h.