Is this endgame drawn after the rook sacrifice?

Question

2R5/8/8/8/4Pp1P/3k1K2/2p2P2/1r6 w - - 0 46

In this position, the engine recommends 1. Rxc2 and evaluates the position as drawn. Is there a general rule in such positions or precise calculation is required to reach such a conclusion?

Answer 1

Let's note the following: 1.Rxc2 and 1.Kxf4 and 1.Rd8+ are equally good, since they all draw. If we ignore 1.Rd8+ for now, 1.Rxc2 is clearly at least as good as 1.Kxf4 in the position, since it drives black's king back one square, meaning that black will at some point have to 'waste' a move with the king to get to white's remaining pawns (the option of 1.Rd8+ introduces a different way for black to play for a win that ultimately doesn't work, but needs to be taken into consideration. This line should be considered if we come to the conclusion that 1.Rxc2 loses by force).

So we begin by calculating a few steps ahead in the 1.Rxc2 line (since it's a direct capture, and if it works then we're happy): 1...Kxc2 2.Kxf4 Kd3 (this is why 1.Kxf4 and 1.Rxc2 are equally good; it doesn't matter if black's rook is on c1 or on b1).

This will be the main tabya of the endgame, and at this point white needs to make a decision: which pawn should be used as the main passer?

Let's consider our options, beginning with the h-pawn:

[White "NN"]
[Black "NN"]
[FEN "8/8/8/8/4PK1P/3k4/5P2/1r6 w - - 0 3"]

3.h5?? Rh1 4.Kg5 Kxe4 5.f4 (5.Kg6 Ke5 {White's king is chained to the h-pawn.} 6.h6 Ke6 {Opposition. Now Black threatens to play ...Rg1+, forcing White's king onto the h-file, and granting access to the f-file for Black's king.} 7.h7 Rg1+ 8.Kh6 Kf7 9.h8=N+ Kf6 10.Kh7 Rg7+! {All that is left to ensure victory for Black is to round up the final pawn.} 11.Kh6 Rg2! 12.Kh7 Rxf2-+ {With the knight in the corner, White is helpless.}) Kd5! {The way black wins in this position is to maneouver their king to the f-file, where it will be able to contain both of white's passers. Since the black rook keeps White's pawns from advancing too quickly, Black will be able to interfere with the king in time.} (5...Rg1+ 6.Kf6 Kxf4?? 7.h6= {and White's pawn promotes.}) 6.Kg6 Ke6 {Black is now threatening to play ...Rg1+, after which Black's king reaches the f-file} 7.f5+ Ke7 8.f6+ Kf8-+ {Black will now be able to either pick off White's two pawns, or White's king will be caught in a mating net.}

So the h-pawn is a no-go. What about the e-pawn?

[White "NN"]
[Black "NN"]
[FEN "8/8/8/8/4PK1P/3k4/5P2/1r6 w - - 0 3"]

3.e5?? Kd4! {The problem with the e5 push is that it allows Black's king easy access to the d5 square, which makes it impossible for White to gain enough time for the pawns to advance successfully.} 4.Kf5 (4.e6 Kd5 5.e7 Re1 {just drops the pawn.}) Kd5 5.f4 Rh1! {luring White's king away from the e6 square, creating access for Black's king.} 6.Kg5 Ke6-+ {Again, White will not be able to stop Black's king from getting to the f-file, after which Black will either mate White's king or pick up the passed pawns one by one.}

Ok, only one attempt left to consider then!

[White "NN"]
[Black "NN"]
[FEN "8/8/8/8/4PK1P/3k4/5P2/1r6 w - - 0 3"]

3.Kf5! {This is the only way to draw, and it is by far the best move in the   position. White waits with pushing a pawn forward until after Black has made a  commitment which leaves White with a tempo to spare in comparison with many of the previous lines, i.e. the answer to the question of which pawn to push depends on which pawn Black decides to go after first!} 3...Re1 {It's not clear which line should be considered the main line for Black in the position.} (3...Rh1 4.Ke5! {This time the f-pawn is used} 4...Rxh4 5.f4 Ke3 6.f5 Rxe4+ 7.Kd6= {There is no way for black to round up the f-pawn.}) (3...Kd4 {The last line we will consider here. Other tries make little sense.} 4.h5= {is essentially a repeat of the main line. The other way to draw is to push the f-pawn forward, but this move is the simplest.})  4.h5! {By waiting with the push, White managed to get Black to waste a tempo on stopping the "wrong" pawn, so to speak. Now there is no way for Black to stop the h-pawn in time.} 4...Rh1 5.Kg6 Kxe4 6.h6 Ke5 7.h7=

Ok, so it turns out that White will be able to draw after 1.Rxc2. Although some precise calculation was needed to make sure that White manages to create sufficient counterplay with the pawns, we have to consider far from every possible line.

This is a common theme for practially any endgame: precise calculation is crucial, but the use of certain abstract "guideposts" in the position (such as "Black's king getting to the f-file = White is doomed" in this case) means that we only have to make a relatively small number of precise calculations to reach the correct conclusion. In general, the more endgame theory you know, the more of these abstract "guideposts" will be visible to you, making endgame calculations far more efficient than brute-foring them.

Answer 2

After Rxc2 Kxc2 Kxf4 we reach the below position, which is a tablebase draw.

The plan for White is really simple: queen the h-pawn (White can give up the rest of the pawns).

Black king is too far away to help, that is why it is a draw (there are winning techniques for Black, but they require kings to be in close proximity).

Feel free to test my claims with the online tablebase. I will offer just one sample line below.

[StartFlipped "0"]
[fen "8/8/8/8/4PK1P/8/2k2P2/1r6 b - - 0 1"]

1...Rg1 {Trying to cut off the White king} 2. h5 Kd3 3. h6 Rh1 4. Kg5 Kxe4 5. Kg6 Ke5 6. h7 Ke6 7. Kg7 {Black must give up the rook, after which Black king picks up the f-pawn } 1/2-1/2

Answer 3

I add my answer too here. (The present answers do an absolutely correct analysis, but methinks the asker is better off with general principles.)

First of all, 1.Rxc2 Kxc2 2.Kxf4 Kd3 and 1.Kxf4 c1Q 2.Rxc1 Rxc1 look effectively like transposition, so I would really have been astonished if it would have made a difference on which file the rook is. But since the black passer costs you the rook anyway, why not have it behind you and deflect the king.

Second, in an endgame KR/KPP... its more or less all about kings. (Look up "shouldering" or "bodycheck" in an endgame book.) The bK is uncomfortably close to the pawns and it doesn't surprise me at all White has to find a lot of only moves. His prime priority is to keep the bK as far away as possible. To find Kf5!, you either must gave extremely good endgame instincts (like me :-) or calculate the whole line. Note that Black has the following advantage in this endgame type: To gang up on a pawn, he just needs to put his R behind it and run up with his K. White needs double as many moves (P and K must advance).

Third, since the wK must advance anyway to assist the P, it doesn't surprise either that Kf5 draws, where e5 or h5 loses as Black has a much better K position in comparison. But generally, you should set your hopes on the h pawn, since a) it's on the far side of the bK and b) stalemate motives.

Fourth, calculate, calculate, calculate. (Or use the tablebase, but it doesn't tell you the why.) But to find a candidate move, use general principles. Luckily, in an endgame there are not that many sensible moves anyway.

Answer 4

It is very close and complex.
I have won many end games with 2ps vs rook. here the black king is out of play and white has 3 pawns black will have to work hard to get the draw. I doubt any computer can analyze this far enough to give a conclusive answer and it was drawn as far as they could see ahead.