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1. Kf5 Kh7
2. Rh1+ Kg8
3. Kg6 Kf8
4. Rf1+ Ke7
5. Rf7+ Kd6
6. Rxg7 Kc5
(it’s a win for white)
As usual, you learn more from what fails- the move white has to avoid making here is the the natural looking
1.Ke6:
1. Ke6?
From a symmetry point of view in regards to the black men, Ke6 and Kf5 are identical- they are reflections on either side of the a1/h8 diagonal that both black pieces occupy, but the moves are not identical with regards to the white rook which only has the f-file cut off from the black king. Let’s see why this matters:
1. ………….Kg8!
The only move that draws for black. The only important alternative is 1. …Kh7, but this allows white to correct his first mistake by playing 2.Kf7! to win analogously as Alena shows above- the black king can’t escape with 2. ..Kh6 since white will check from h1 and then skewer the black pieces with Rg1+ winning the bishop. However, by putting the king on g8 while the white king is on e6, black foils white’s attempts to mount deadly mating threats on the edge. Now white needs to get the king to g6 to threaten black, but doing so causes problems….
2. Kf5 Kf7
And the black king escapes the edge and has a theoretical draw because the white king can’t cross back over the f-file without opening that file to the black king by cutting the rook’s line into f8/f7. Additionally, at move 2, it will do white no good to try to wait black out with another king move or rook move:
2. Ke7 Bb2
Black draws with any move any move along this diagonal except for the obviously bad Ba1, but also the losing Bh8 which takes a critical square away from the black king in a critical line: [2. ….Bh8?? 3.Rg1 Bg7 (forced-see below) 4.Rg2 Kh7 5.Kf7+-]. Continuing from Bb2 above:
3. Rg1 Kh7!
4. Kf7 Kh6 (the critical square that 2. ….Bh8 takes away- the king gets skewered in the analogous line)
And the black king escapes because there is no immediate mate threat, and the black bishop controls f6.
By getting the white king on the f-file before the black king can get to g8, white lets the rook prevent the escape in that direction while the king and the threats of Rh1 prevent the escape in the other dimension.