Is that really true, Alena? Compare the two solutions you offered- the first incorrect one, and then the second and correct one. Then consider the line you omit that is important to finish this out: 5....Rxa3.

The king controls the important square e3. If the king moves onto c4 instead of d4 it will lose a critical tempo.

And why does 4.Kd4 win and 4.Kc4 only draw?

Yancey, the problem slipped from my mind. Now you reminded me about it.

1.Bh6 Kxh7
2.Bxg7 Rb8
3.e6 Rb3+
4.Kd4 Kg6
5.Kc4 Rb1
6.g4 Rb3
7.g5 Rxa3
8.Bh6 Rf3
9.e7 Re3
10.f7 Kxf7
11.g6+ Kxe7
12.Bxe3 Kf6
13.Kb4 Kxg6
14.Kxa4 Kf5
15.Kb5 Ke4
16.Kc6 Ke5
17.Bg5 Ke4
18.Bd8 Ke5
19.Bxc7 Ke4
20.Kxd6 Kd4
21.Ke6
It's a winning position for white.

Have you made any progress, Alena?

Yes, you are right, I did find out that white can't escape the queen checks after 6....Rf3

In other words, why was 6.....Rf3 possible, and why did it fail? You have probably already determined that the problem for white in that line is that white can't escape the queen checks in the key line after 6....Rf3, right?

The issues are very subtle in this problem, Alena. As I wrote in the introduction- the theme is zugzwang. Because of the threat of the white pawns, black must try to generate counterplay with his a-pawn while, at the same time getting the rook into position behind the white pawns on the kingside. To foil this, white has to choose his moves in right order so that black is forced to make his moves at the wrong time.

1. Bh6 Kh7 2. Bg7 seems like a win for white
   eg
2. Bh6 Kh7 2. Bg7 Rg8 3. f7 Kg7 4 fg8Q Kg8 e6 and the e/g pawns plus WK look too much for BK

Yancey, I couldn't find a win for white after your 6...Rf3.
I will try to find a mistake before the sixth move.

Alena, what about 6.....Rf3

V-1

1.Bh6 Kxh7
2.e6 Kg6
3.Bxg7 Rb8
4.Kc4 Rb3
5.g4 Rxa3
6.g5 Re3
7.Bh6 a3
8.f7 Rf3
9.f8=Q Rxf8
10.Bxf8 a2
11.e7 Kf7
12.e8=Q+ Kxe8
13.Bg7
It's a winning position for white.

1. Bh6 gh6 2. e6 Rb8 3. f7 Rb3+ 4. Kc2 Kg7 5. h8Q+ Kh8 6. f8Q+

1. Bh6 gh6 2. e6 Rf6 3. e7