Black to move. Is this a win or draw? How should black proceed?
Source: ChessToday.net
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Black to move. Is this a win or draw? How should black proceed?
Source: ChessToday.net
1. b2!
2.Kxb2 Kb4
3.Kc2 Ka3 collect the d pawns and win.
if:
1. b2!
2.Kc2 b1=Q!
3.Kxb1 Kb3–>Kc3 collect the pawns and easy win.
Black wins easily, through triangulation.
First principle: White cannot move Kb2 if Black can respond with … Kb4; with White forced to move away, the Black king will infiltrate through c3.
Second principle: If Black plays 1. … Ka3 while White’s king is on b1, White responds with 2. Ka1 and Black has not made progress, since 2. … b2+; 3. Kb1, Kb3 is stalemate.
Third principle: If Black plays 1. … Ka3 while White’s king is on a1 or c1, Black wins after 2. Kb1, b2; 3. Kc2, Ka2.
Black uses triangulation to get to the position shown, but with White to move:
1. … Ka5
2. Ka1
2. Kb2, Kb4 wins for Black – see first principle. 2. Kc1, Kb5 is similar to the main continuation, since after 3. Kd2, Kb4 Black would infiltrate.
2. … Kb5
3. Kb1
Again, 3. Kb2, Kb4 wins for Black.
3. … Ka4
and now we’ve repeated the position but with White to move. Zugzwang as anything White plays results in Black infiltration:
4. Kb2, Kb4 (first principle)
or
4. Ka1 (or Kc1), Ka3; 5. Kb1, b2; 6. Kc2, Ka2 (third principle).
Anytime one King’s movements are restricted (here, by the Black pawn on b3 which prevent Ka2 or Kc2), triangulation is probably the answer.
The players are now in mutual zugzwang. The key is who is to move!
With Black to move it is a draw as white has the opposition!
>A-1…Ka3.2.Ka1.b2+.3.Kb1.Kb3. Draw. Both Black and white are forced to draw
>B-1…Kb4.2.Kb2.Ka4.3.Kb1 and black cannot make any headway and it is a draw.
However if it is White to move then black has the opposition and wins!
>A-1.Ka1.Ka3!.2.Kb1.b2!.3.Kc2.Ka2. And 4….b1=Q
>B-1.Kb2.Kb4!
>>B1-2.Ka1.then Ka3 as in A
>>B2-2.Kb1/Kc1.Kc3! and the d pawns fall for a easy win.
In the actual game Black won because it was white to move and white resigned, I think, in the position.
1) …Ka4-a3
2) Kb1-a1 [if White King should move to c1, Black King simply moves to a2 securing b1 and queens the b-pawn]…Ka3-b3
3)Ka1-b1…Kb3-c3 [I would expect White to resign after King c3, recognizing the loss of their pawns on d and e; or as a wise sage would say Black’s advantage is Decisive!]
Black is definitely able to win this game.
I don’t know for certain whether an immediate Ka3 throws the win away, but it surely makes no progress, and its natural continuation is a likely draw:
1. ……Ka3?/?!
2. Ka1
And now I think b2 is going to be a draw…
2. ……b2?
3. Kb1 Ka4 (Kb3 stalemate)
4. Kb2 Kb4
5. Kc2
And, I am not 100% sure this can’t be won now, but I don’t see how black can manage it. Black would have to try to maneuver around to find a way to exchange at e5 to get a won ending, but I can’t see how that works, but I am not going to go that deeply into it since the clear winning method is quite easy for me to see.
The starting position is a clear win for black if it were white’s move. So, black needs to maneuver so that the same starting position is reached, but with white to move. The easiest way to accomplish this is to back the king up on the first move. Black will take advantage of the fact that white not only needs to guard against the b-pawn, but must also keep the black king out of c3.
1. ……Kb5 (Ka5 should be similar)
2. Kc1
Or if white plays 2.Kb2, black takes the opposition with Kb4 and will play Kc3 on the next move to win the doubled white pawns and the game along with them. Continuing:
2. ……Ka5
Keeping the king out of a4 for the moment. White still can’t play Kb2. Now, from here, there are two main lines, and I will cover them both in order:
3. Kb1 Ka4
And black has achieved the initial objective- reached the same starting position, but with white to move. This is lost for white:
4. Ka1
Again, 4.Kb2 is met by Kb4 followed by Kc3. 4.Kc1 is met by Ka3 and b2 as in the line that follows. Continuing:
4. ……Ka3
5. Kb1 b2
And the black king will move to a2 and the pawn will queen afterwards.
Finally, at move 3 in the line above, the white king could have moved to the d-file, but this, too, is lost:
1. ……Kb5
2. Kc1 Ka5
3. Kd1 Kb4 (threatening Kc3)
4. Kd2 b2 (Ka3 ok, too)
5. Kc2 b1Q
6. Kb1 Kc3 with an easy win from here.
I wrote a post early morning that if it is white to move, black wins and if it is black to move it is a draw
That was bothering me for a long time and I realized that black can win by triangulating for white cannot triangulate due to the pawn covering both a2 and c2 squares.
Yes it is a win for black irrespective of who has the move.
If black had only “b” pawn it would have been a draw as P is ahead of K. However presence of other pawns needs just access to c3 for black K.right now it is not possible because 1….Kb4 is met by 2.Kb2.
The time tested distant opposition helps in this case.
1…. Kb5
(and white is in zwugzwang.
2.Kb2 Kb4 and white has to allow black K to access c3.)
2.Kc1 Ka4
3.Ka1 Ka3
4.Kb1 b2 black wins.
1. … Ka3 wins for black.
Black would like to have this position with White to play. This is easy to achieve
1… Ka5 2. Ka1 Kb5 3. Kb1 Ka4
(If white elects to play Kb2 at either turn, black responds …Kb4 and gets the K to c3 on his next turn.)
Now white has the choice of
A. 4. Ka1 (or Kc1) Ka3 5. Kb1 b2 6. Cc2 Ka2 and queens
B. 4. Kb2 Kb4 and black gets to c3 next turn with an easy win by trading the b-pawn for the two d-pawns.
To try to tie up loose threads from my earlier comment, I took a longer look at the line where black plays an immediate Ka3. When I wrote my first comment, I wasn’t sure if black tosses winning chances away on the first move of Ka3. I can now answer that question with certainty:
1. ……Ka3
2. Ka1 Kb4 (for b2 see below)
3. Kb2 Ka4
And now white has no good options. Kc3 is met by black playing Ka3 followed by Ka2-b2-b1Q. If white plays either Ka1 or Kc1, black then plays Ka3 followed by b2 to win as I described in my first comment. Finally, if white plays Kb1, we have returned to the beginning position where black can win by retreating the king to the fifth rank in order to force white into zugzwang.
Finally, I was unsure if black has tossed winning chances away on the following line where he has played both Ka3 and b2 on the first two moves. I haven’t really come to conclusion here, even with a longer, deeper look. The main plan I see for black is to force the exchange at e5, and then trade the d-pawn for the g and h pawns
1. ……Ka3
2. Ka1 b2
3. Kb1 Kb4
4. Kb2 Kb5
5. Kc3
Not sure what white does here that is better. Facing the king off by moving via 5.Ka3-6.Kb4 etc. allows black to exchange at e5 and then force a trade of the d-pawn for the white pawns on g4 and h5. That ending might well be lost for white. Rather than play out those lines (tedious), I just put into a Nalimov tablebase some positions that arise where the pawns at d4 and e6 have been exchanged, the white king has captured the remaining black d-pawn at either d5 or after being pushed to d4, and that the black king has captured at g4 already- those positions are lost according to the Nalimov tablebase (you will see them again, below in some of the variations). From 5.Kc3 above, I intended to try to find a line where white isn’t forced to make those trades. Continuing the main line:
5. ……Kc6
6. Kd2 e5
7. de5 d4
8. e6 Kd6
Again, a critical juncture, with uncertainty. Black has the upper hand here. I am going to play out a few of the lines from this point, so bear with me:
9. e7 Ke7
10.Kc2 Ke6
11.Kb3 Kd5
12.Kb4 Ke5
13.Kc4 Kf4
14.Kd4 Kg4
And, this is exactly one of the positions I put into the Nalimov Tablebase, and it tells me white wins in another 24 moves. So, let’s back up to move 10 for white. Can he successfully prevent the trade of the d-pawn for the g and h pawns?
10.Ke2 Ke6
11.Kf3 Ke5
12.Kg3 Kd6
13.Kf3 Kd5
14.Ke2 Kc5
15.Kf3 Kb4!
16.Ke4 Kc3
And the d-pawn will fall, and white doesn’t have enough time to win at h6 and g5 to queen his own pawn in time:
17.Kf5 Kd3
18.Kg6 Ke3
19.Kh6 d3!
20.Kg5 d2
21.h6 d1Q
22.h7 Qd4
23.Kg6 Qh8
24.Kh6 Qf6! and mate follows in another two moves.
So, you can see the difficulty. I don’t know that white can ever save the draw, even if black plays imprecisley at the start. There are a lot of variations in the line above that I didn’t look at.
I think Bob nailed it and pointed out the key concept of triangulation. I tried the concept of distant opposition with 1… Kb5 but got stumped after 2.Kc1 when Black couldn’t keep the distant opposition since White’s pawns prevent …Kc5. Sigh… it’s a K+P ending. I’ve played chess long enough to be able to figure these out , right?! 🙁
Craigaroo
@Ben
In your line [1…b2], after 3…Ka3, nothing stops White from playing 4.Kc3 (opposition)and Black hasn’t achieved anything.
– Craigaroo
Craigaroo,
The moves 1. …Ka5 and 1. …Kb5 are equivalent:
1. ……Kb5
2. Kc1 Ka5!
And
1. ……Ka5
2. Kc1 Kb5
In both cases, black is waiting for the white king to return to b1 before playing Ka4 again.
And you can follow the rest for Kb5 from my comment above.
I think, sometimes, players can get hung up by trying to follow concepts such as opposition and distant opposition too literally. The idea is a bit more complex than just being able to reply to white’s Kc1 with Kc5, for example, and this problem illustrates that since the black king on a5 with the white king on c1 with white to move is an effective demonstration of the concept of diagonal opposition- the idea being that the black king stands ready to come to b4 if white plays to b2 (the standard idea of opposition).
There was a problem several years back posted here where the full solution involved a deep discussion of the various ideas of opposition. If I can find it, I will post the link here if you are interested.
Yancey said: “The moves 1. …Ka5 and 1. …Kb5 are equivalent…”
Yeah, I realized that much after some of you showed the solution beginning with 1…Ka5 but I just totally forgot about triangulation. So no points to me for starting with the distant opposition. Thanks for the comment and if you want to post that link about opposition, yes I would look at that. Thank you!
– Craigaroo
Craigaroo,
Well, you do get points for starting off with the idea. The entire point of backing up the king to b5 is the establishment of distant opposition- waiting for white to make the fatal move that allows direct opposition. My only point is that outside direct opposition, the entire idea is more complex than the kings opposing each other on a file or rank across three squares. This problem makes that point- Ka5 also establishes an effective distant opposition- caused by the intruding black pawn’s control over a2.
In any case, here is the link. There are multiple comments by myself and Cortex as we hashed out the issues involved- at each point, Cortex was pressing new concepts about the problem. I probably learned more about king and pawn endings from this one problem than any I ever studied.
Link