Only one winning move here- push the pawn. This is a common theme in these kind of endings- the white king is vulnerable to a skewer if white takes on the seventh rank (also on the 8th rank in some variations):
1. …..h2!
Any other move will draw as white will play Ke3 on Ra2+, and will play Ke1 if black plays Ke5. The threatened skewer is only effective on this first move. The rest is straightforward walking down the rook to stop the checks:
2. Rh5
Nothing can hold. If white moves the king to e2 or c2 or moves the rook to a non-checking square on the h-file, black plays Ra1 which forces white to capture at h2 (now or later after black walks the rook down on the h-file) and then black skewers from a2 to win the rook. If white tries Ke1/d1, black just queens and then skewers from a1 instead. The white rook can check from the h-file for a while, but the black king walks him down. Continuing from 2.Rh5 above:
Of course, 6.Rh2 is met by Ra2+ as mentioned earlier. Continuing:
6. …..Kf5
And white can keep checking until the black king reaches the 7th rank at which time the rook will have to give itself up to stop the queening, or play R vs R+Q.
1. Ke5 Rh5+
2. Kf4 Rh4+
3. Kg5 1-0
Mark
Only one winning move here- push the pawn. This is a common theme in these kind of endings- the white king is vulnerable to a skewer if white takes on the seventh rank (also on the 8th rank in some variations):
1. …..h2!
Any other move will draw as white will play Ke3 on Ra2+, and will play Ke1 if black plays Ke5. The threatened skewer is only effective on this first move. The rest is straightforward walking down the rook to stop the checks:
2. Rh5
Nothing can hold. If white moves the king to e2 or c2 or moves the rook to a non-checking square on the h-file, black plays Ra1 which forces white to capture at h2 (now or later after black walks the rook down on the h-file) and then black skewers from a2 to win the rook. If white tries Ke1/d1, black just queens and then skewers from a1 instead. The white rook can check from the h-file for a while, but the black king walks him down. Continuing from 2.Rh5 above:
2. …..Ke4
3. Rh4 Kf5
4. Rh5 Kg4
5. Rh8 Ra1 (Rh3 also wins)
6. Rg8
Of course, 6.Rh2 is met by Ra2+ as mentioned earlier. Continuing:
6. …..Kf5
And white can keep checking until the black king reaches the 7th rank at which time the rook will have to give itself up to stop the queening, or play R vs R+Q.
1…. h2 wins.
1…. h2
2. Rh5+ Ke6
3. Rh6+ Kf7
4. Rh7+ Kg6 (checks finished)
5. Rh8 Kg7
6. Rh5 Ra1 (forcing)
7. Ke3 trying to get out(other alternatives like 7. Rg5+ Kh6 and white will loose track of h file.
7…. h1Q
Black wins.
1…h2 -+
hmm:
1. …h2
2. Ke2(or c2) Ra1
3. Rxh2 Ra2+
h2 Rh5+ Ke4 Rh4+ Kf5 Rh5+ Kg4 Rh8 Kg3 Rg8+ Kf2 Rf8+ Kg2 Rg8+ Rg3! (if Rxg3 Kxg3 Ke2 and h1Q) so Rh8 h1Q is full queen up