Is this endgame position a draw?

Question

It’s white to move.

[FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]

I was white, and this was a blitz game which ended up as a draw because I didn't know what to do. My friend and I later analyzed this position, and he suggested that if I had played Rh3! here, then I would've won (of course, if black trades rooks, white wins). But I wasn't convinced, as the trade wasn't forced at all, and judging by the black king's position, I thought black could hold with perfect play (i.e. keep his king where he is, and use rook checks/occupation of the f-file to stop the white king from getting too close to the white pawns). We gave up analyzing after a while, and used an engine to check the position, and it gave +3 to white, which would imply (I think) that there is a forced win. But machines still tend to be wrong, and after my friend played out the line given by Fritz 12, no progress was made by white. The black rook simply occupied the f-file, and the white king couldn't get close to the white pawns.

Is this a draw? If this isn't a draw, how can white win?

Answer 1

Yes, it seems to be drawn. First consider the following variation, but in particular the position after 2.Kxh5 first:

[FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]

1.Rh3 Rg1 2.Kxh5 Rg8 3.Rf3 Kc4 4.Kh4 Kd5? 5.Rf5+!

Here we don't even need to consult an engine because there is a better oracle to be had. This 6-piece position is covered by existing endgame tablebases, and you can confirm with a query that this is drawn. So we know for a fact that going this 1.Rh3 route leads to a draw. (White could try delaying the pawn capture and making a more productive immediate move, but it doesn't help, and just leads to play like that coming after 1.Kg5, discussed at the end below.)

As to the idea behind Black's play from here, it is to keep the pawns at bay with the king, and as you intimated in your original post, to use the rook to cutoff the white king as much as possible. It's difficult to say much more about the exact drawing method here, because the play can be very complicated (one reason to love endgames, by the way). To give an idea, consider the position after 4.Kh4. 10 of Black's possible moves keep the draw (including those that simply move the rook along the g-file), but the perhaps seemingly harmless 4...Kd5?, putting the king back where it was, is suddenly a loss for Black in 41 moves, but only if White plays 5.Rf5+!, as all other fifth moves again draw. One way to get a good feel for the correct and incorrect play (and for the general notion of how impenetrable optimal play in technical endgames can be) is to toy around with moves at the link I gave above.

White's only other first move option is 1.Kg5, and it's not too surprising that this draws if winning the h-pawn immediately already draws. It's true that White avoids getting his king cut off on the h-file, but the cost of leaving Black with a distracting outside passed pawn is too high. The point is that the black king and rook can easily stop the pawns if the white king isn't helping out his rook. But now if the white king does try to get involved, say via ideas like Kf6-e6 and pushing the d-pawn, Black can keep things level because she has her own passed pawn to play with.

Answer 2

While the accepted answer gives the correct evaluation of the position and a useful link, I think it is important to detail a little bit more the strategy in this interesting rook endgame.

First of all, it is important to notice that Black cannot keep the white king cut away from the pawns. Offering rook exchanges and hiding behind the white rook, the king will make its way until d2.

 [FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]

 1.Rh3 Rg1 2.Kxh5 Rg8 3.Rf3 Kc4 4.Kh4 Rg7 5.Rg3! Rf7 6.Kg4 Rf8 7.Rf3 Rg8 8.Kf4 Rf8! 9.Kg3 (9.Ke3 Re8)  (9.Ke4 Re8! {and the king has to go back}) Rg8 10.Kf2 Re8 11.Re3 Rf8 12.Ke2 Rd8 13.Kd2

Actually, if the white king could rather reach the upper part of the board in front of its pawns (say, e5 or d6), he would win, but as shown in the sub-variation above, Black can prevent this with frontal checks since the white rook has to stay on the third rank protecting its pawn on c3.

When the white king reaches d2, White has a strong threat of Re5-c5+, kicking the black king out of its blockading position. To save the draw, Black must react and prepare a check on the second rank, grab the Pc3 and reach a well-known KRPKR theoretical endgame:

 [FEN "3r4/8/8/8/2kP4/2P1R3/3K4/8 b - - 0 1"]

 1...Rh8! 2.Re5 Rh2! 3.Ke3 (3.Re2 Rh3) Kxc3 4.Rc5 Kb4 5.Rc8 Rh3 6.Ke4 Rh4 7.Ke5 Rh5 8.Kf4 Rh4 9.Ke5 Rh5 10.Kf6 Rd5

After 4.Rc5 the black king is cut one file away from the white pawn and White is aiming for a winning Lucena position. There are two possible defenses for Black:

a. frontal checks, which works when the pawn has not crossed the half of the board: Re8-d8-e8 etc harrassing the white king. White can prevent that with 5.Rc8!?

b: lateral checks, which works when the 'long side' of the pawn is available, i.e. when the defensive king is cut on the 'short side': it is the case here, so black can secure the draw by checking from the h-file 5...Rh3 6.Ke4 Rh4 7.Ke5 Rh5 etc. White cannot make any progress.

Thus the king being cut on the short side is the decisive factor. Indeed, if in the starting position we move the central pawns one file to the right, White is totally winning:

 [FEN "8/8/8/4k2p/4P2K/3P2R1/8/7r w - - 0 1"]

 1.Rh3 Rg1 2.Kxh5 Rg8 3.Rf3 Kd4 4.Kh4 Rg7 5.Rg3! Rf7 6.Kg4 Rf8 7.Rf3 Rg8 8.Kf4 Rf8 9.Kg3 Rg8 10.Kf2 Re8 11.Ke2 Rh8 12.Rf5! Rh2 13.Kf3 Kxd3 14.Rd5 Kc4 15.Rd8 Rh5 16.Kg4 {The black rook lacks a i-file.} Re5 17.Kf4 Rh5 18.e5 {White progresses and will reach a Lucena position soon.}

Answer 3

This is a very late answer, but if you check 7-piece Lomonosov tablebases, it shows the original position is a draw.

Here is a link to the proof.