The tablebase suggests it's a win for White, but I struggle to beat my phone. Black's rook seems to always be able to harass the white king to stop white make any progress.
What's the technique to win this endgame?
[fen "8/8/6k1/6r1/8/8/6PP/5RK1 w - - 1 61"]
I remember drawing a rook and two pawns vs. rook position with g- and h- pawns. I was the losing side. Basically, 1) prevent king mobility, 2) win the rook pawn and 3) play the first rank defense on the g-pawn.
In rook endings, there are many positions where two pawns are not enough to win. Only opposite-coloured bishop endings present a higher drawish trend.
Excerpt From: "100 Endgames You Must Know: Vital Lessons for Every Chess Player Improved and Expanded" by Jesus de la Villa.
But what if the opponent plays accurately, and doesn't drop pawns?
The idea is as follows: set a blockade on the seventh rank by playing Rb7 and Kh7. If the king approaches, check him with your king on g6. If the rook attacks your king on h8, White has no time to move his King forward on the next move because of ...Kxg8.
To win, prevent the opponent from enacting this blockading strategy. Judging by the original poster's position, Black plays 1...Ra5 (or 1.Ra1 Rb5) and blockades.
Draw.
EDIT It is a draw only if the pawns are pushed without thought as another poster suggested. The strategy is to push slowly, one at a time and to avoid drawing positions as listed.
A two pawn advantage in most cases guarantees the win. But there are quite a few drawn positions, particularly when one of the pawns is a RP (as in OP).
Excerpt from Rook Endings. Smyslov and Levenfish.
This book is a great reference on such positions. There are eight pages devoted to it. Here is an excerpt:
original post, black plays 1...Ra5 and blockade, draw, do you mean the board position in the question? 6-men database says it's win for white no matter who's on the move.
As we can see from the other answers, there are two things you should avoid: 1) The pawns getting blockaded. 2) Your king getting cut off from the pawns.
It is not the only way to win, but as a matter of technique it is a good thing to advance the h-pawn before the g-pawn. In this way your king can use the g-pawn to hide both from checks from the side and checks form behind. But first use the rook to chase the opponent king as far back as possible.
Example:
[FEN "8/8/7k/R7/6PP/6K1/8/1r6 w - - 0 1"]
[Title "Rook endgame g+h"]
1. Ra6+ Kg7 2. h5 Rb3+ (2... Rg1+ 3. Kh4 Rh1+ 4. Kg5) 3. Kh4
The checks will stop and you can think of the next step.
[FEN "8/6k1/R7/7P/6PK/8/8/1r6 w - - 0 1"]
[Title "Rook endgame g+h"]
1. Ra7+ Kg8 (1... Kh6?? 2. g5#) (1... Kf6 2. g5+ Kf5 3. Rf7+ Ke6 4. Rf6+ Ke7 5. h6) 2. h6 Rb5 3. g5 Rb1 4. Kh5
Here I would again chase the king away before advancing the h-pawn another step.
If the g-pawn is pinned there is no point in giving a check first.
[FEN "8/7k/R7/7P/1r4PK/8/8/8 w - - 0 1"]
[Title "Rook endgame g+h"]
1. h6 (1.Ra7+ Kh6) Rb1 (1... Rb5 2. g5 Rb1 3. Ra7+ Kg8 (3... Kg6 4. Rg7+ Kf5 5. h7))2. Kh5 Rb5+ (2... Rh1+ 3. Kg5) 3. g5
In this way you can advance slowly rank by rank. Towards the end black will have to defend passively to avoid mate at the back rank. Just for the fun of it I would try to end the game with a knight or bishop promotion like in my answer to this question.
[FEN "r6k/8/6PP/5RK1/8/8/8/8 w - - 0 1"]
[Title "Rook endgame g+h"]
1.Rf8+ Rxf8 2.g7+ Kh7 (2...Kg8 3.Kg6) 3.gxf8=N+!(3.gxf8=B!)(3.gxf8=Q??)(3.gxf8=R??)
Seems trivial, push pawns and use rook to block. I'm not sure what part you would have trouble at. There isn't even a general technique for this because it's a clear cut win for white.
You need to post the position where you have trouble winning in, such as avoiding stalemate traps. It is hard to give a solution from this position.
The following process can be used:
1) Push the h-pawn and put your king on the h-file. The g-pawn is used as a shield from the black rook's horizontal checks.
2) Put your rook on the queenside, on either the rank the g-pawn is on right now, or on the rank above.
3) Push your g-pawn one, followed by advancing your king up one square. The picture on the kingisde looks something like Kh3, pawn on g3, and pawn on h4.
4) If Black's king is directly opposing you (say on the h5-square), check it with your rook. If Black's rook is preventing this (e.g., by being on the a5-square), then nothing is stopping you from advancing your g-pawn with g4+.
5) Then, advance the h4-pawn to h5. Now your king has a place to hide on h4 (since your g-pawn is either ready to blockade on g4, or it's about to advance to g4).
Repeat this general process. Essentially, before advancing a pawn, make sure your king can either move up and get new shelter, or that your rook is already on the same rank to prevent the king from being checked in the first place.
