6
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?

9

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.

| improve this answer | |
  • Why not simply 5. Kg6 in your first line rather than pushing the f pawn? It seems like Black is just a square behind and I don't really see what f4 gains White there. – Steven Stadnicki Jul 4 at 19:39
  • 1
    @StevenStadnicki Black will run their toward e6 immediately then. After this, Black will be able to push White's king away from the g-file with ...Rg1+, after which it has to step onto the h-file (which renders the threat that White's h-pawn poses useless). If White tries to prevent Black from pushing the king away, Black's king will reach f8, neutralizing White's counterplay. I'll add one variation above to illustrate what the problem is. – Scounged Jul 4 at 19:47
6

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
| improve this answer | |
-5

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 analyse this far enough to give a conclusive answer and it was drawn as far as they could see ahead.

| improve this answer | |
  • 3
    As you can see in the other answers here, it is possible to prove that the game is drawn with best play. As noted in one of them we have tablebases to confirm this, and as I noted in my own answer there is a very concrete way for White to draw the position assuming best play from both sides. If you feel like the answers that have already been provided are lacking in some regard, feel free to fill in the gaps. But as your answer stands it adds nothing of value (on top of being plain wrong. Top engines nowadays manage to find optimal play in this position with ease.) – Scounged Jul 4 at 20:04

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