In the fun Kasparov Endgame thread (Link), Ajezz mentioned corresponding squares and I'm sure there are a few posters here who might be a little confused by this phrase. I know it took me quite a few years of playing before I found a clear explanation. So here's an introduction that I hope will help out newer players or anyone who simply hasn't gotten a good explanation before.

Definition

Dvoretsky defines Corresponding Squares as "squares of reciprocal zugzwang". Awesome, but what does that mean? "Zugzwang" describes a situation where one player is put at a disadvantage because he has to make a move. i.e. passing your move would be a better option than moving. "Reciprocal zugzwang" means that whichever side to move is at a disadvantage because they have to move. This may sound esoteric or confusing, but it'll make sense after we see a few examples.

Dvoretsky also helps us out tremendously by listing the 3 most commonly seen cases of corresponding squares: opposition, mined squares and triangulation.

Opposition



This is an example of close or simple opposition. The kings are on the same file and have one square between them. Whichever king is to move does NOT have the opposition. The opposition is key to this position. If white has the opposition (black is to move), then white can win. If black has the opposition (white is to move), then it's a draw. Great, but how does this relate to corresponding squares? In this case e5 and e7 are corresponding squares. They are squares of reciprocal zugzwang because whichever side is to move is at a disadvantage.

Mined Squares

Okay, you've probably heard of opposition before, but what are mined squares? Mined squares are squares where you must not place your king before you opponent does. Again, if you place your king on one of these squares, your opponent will be able to immediately place you in zugzwang.



The two question mark squares are "mined". If either king steps onto one of them, then the opposing king can step onto the other and immediately win the enemy pawn. These squares are corresponding because if both kings are on those squares, then whoever is to move will lose.



What are the two "mined" squares here?

Triangulation

Triangulation relies on understanding which squares are corresponding. Let's look at a classic example to see how this works.



Here it's white to move. Can white immediately make progress from this position? No. If we have this same position, but with black to move, can white make progress? Absolutely. After 1 .. Kc8 2. Kb6 white wins easily. So what can we say about c5 and c7? They are corresponding squares. With the kings on these squares, you'd prefer your opponent have to move. Okay, are there other corresponding squares? Let's look at another possible position for the kings.



Again, can white immediately make progress if he is to move? No. 1. c7+ Kc8 2. Kc6 drawn. If black is to move, can white make progress? Absolutely 1 ... Kc8/Ke8 2. c7 and white wins. So we have two more corresponding squares: d6 and d8. So we should start thinking about the position like this:



The two A squares correspond and the two B squares correspond. Are there more corresponding squares? Absolutely. From d5 and c8, the kings can go to either of the corresponding squares that we've already established. So d5 and c8 also correspond. Then you figure out where the kings can threaten to go to C and A from and you end up with this:



So from the initial position, the winning line goes 1. Kd5 Kc8 (C for each) 2. Kd4 Kb8 (D) 3. Kc4 and black has no legal corresponding square - b7 is controlled. 3 ... Kc8 4. Kd5 back to the C's only now black has to move 4 ... Kd8 5. Kd6 and white wins. Other tries by black also lose.

Please ask questions or clarify these points if they don't make sense!

nice explanation, I've spent enough time this week drilling people in K+P vs K that maybe I should move on to this

Awesome post.

A+ swingdoc u rule

Thanks for the comments, guys. Here's the quiz for this lesson. These aren't the simplest problems ever, so I'm including a hint or two for each problem. Obv try to solve each problem first and only then look at the hint. The second hint normally gives a lot more away than the first hint.

#1 White to move and win. This is a classic. Find the winning line and 3 pairs of corresponding squares.






#2 White to move and win. Give a winning line and list at least 4 pairs of corresponding squares.

Spoiler:

This problem relies on opposition and the fact that trading the kingside pawns loses for black.

Spoiler:

If white can achieve horizontal opposition (along a rank instead of a file), he will win.

#3 Black to move and draw. First find 5 pairs of corresponding squares and then give the only move that draws for black.





#4 White to move and win. Find 3 pairs of corresponding squares and give a winning line.

Awesome post Swingdoc!

Dvoretsky explains this concept perfectly in his great book "die Endspieluniversität", unfortunately i don't think that's available in English.

I'll try #1

Excellent post!

Thank you so much swingdoc, this is really excellent

Excellent try! You did miss an important defensive resource for black. What happens after 1. Kc4 Ka5 2. Kd4 Kb5?

*White to move


What is the correct evaluation of this position? What would happen in this position if black is to move? What does that mean about this position?

This is one of the critical parts for understanding this whole problem, so please anyone who didn't instantly know how to win this, chime in! For more advanced players, what's the answer to questions 2-4?

Edit - also important for answering this problem is this position:



If white is to move, who wins? What are the "mined squares"?

If it's black to move, then white wins the pawn. That means this is reciprocal zugzwang/ d4 and b5 are corresponding squares.

After 1. Kc4 Ka5 2. Kd4 Kb5, why not 3. a4+ Kxa4 4. Ke5 Kb5 5. Kd6 and white has opposition? <edit>Is it also correct to say that in this position, b5 and d6 are mined?</edit>

In your last example, the mined squares are d4/f5. If white is to move, black wins because after 1. f5 d4 black has opposition and plays 2....Kxe4.

close but...

Close, but you're a little off on this position. Assuming white to move, is there a way for white to force black to step on the mined squares (d4) before white steps on the corresponding square (f5)?

Absolutely amazing post Swingdoc, much appreciated!

When I first glanced at the problems I was under an impression that I should solve them easily, but it turned up to be not that easy, so here's my try, flame away if I missed something for black

1. Pretty sure got this one right


2.


3.



4.

Oh duh.

Yes! And so whoever is to move in this position, wins.



But who cares, right? How does this apply to problem #1? After 1. Kc4 Ka5 2. Kd4 Kb5 3. a4 Kxa4 4. Ke5 we get this position with black to move:



Funny enough, both you and YouKnowWho gave the same wrong move for black here, although his seems to be based on a pretty sophisticated defensive idea that just happens to fail. So what is the right move that allows black to draw?

Not bad. You did miss some of the better defensive tries.

1.

2.
3. Perfect

4.

Well, hopefully this thread has helped some people and just so I don't leave you hanging, here are the answers:

#1


Hopefully the last few posts demonstrated why d4 and b5 correspond. The starting squares for the kings correspond (b4 and a6) and if the a pawn is gone, b5 and d6 correspond. The solutions:

1. Triangulate to "gain" a move
2. Walk our king around the c pawns being careful to AVOID d4 unless black plays Kb5
3. Profit!

1. Ka4 Kb7 2. Kb3 Ka6 3. Kb4 Kb7 4. Kc4 Ka6 5. Kd3 Ka5 6. Ke4 Ka4 (Kb5 7. Kd4) 7. Ke5 Kxa3 8. Kd6 +-

#2


Corresponding squares here rely on the opposition. If white can get the opposition he can force the trade of the f&g pawns. Black's task is complicated by the fact that his king must be inside the black line (3 or fewer moves from capturing the g-pawn) if white threatens the b-pawn.

1. Ke4 Kg4 2. Kd5 Kh5 3. Kc6 Kg4 4. Kd6 Kg5 5. Ke5 Kg4 6. Kf6 Kh4 7. Kf5 Kg3 8. Kg5 +-

#3


This one is fun because figuring out the corresponding squares makes it super easy to defend. The key is that when the white king is on a2, the black king must be on b4 to defend. The rest of the squares are easy once you've figured that out.

1 ... Kf3 =

#4


Easy to see that c3 and e3 might correspond. The starting king positions also make sense as if black has to move, white will infiltrate along the e-file. Now it's like playing minesweeper. C squares are the only ones that allow the kings to threaten both A and X. b3 and b2 allow the white king to threaten A or C and black has no corresponding squares.

1. Kc2 Kf4 2. Kb3 Kf3 3. Kb2 Kf4 4. Kc2 (triangulation!) Ke5 5. Kd1 Kd5 6. Ke2 Kd4 7. Kd2 Ke5 8. Ke3 Kd5 9. d4 +-

Thanks for the thread, swingdoc. I'm sure it took a lot of time, and it was very well done, and an interesting topic.

(I hate to mention it, but the solution to #2 got a little jumbled towards the end, though the idea is already clear by that point. Maybe you can enlist a friendly mod to adjust it, since I'm sure this thread will be linked plenty of time in the future.)

I just noticed this. Is that the same book as his Endgame Manual? That's the first book that clearly explained this idea to me and the source for all of my quiz positions.

Thank you very much! I loved doing this thread because it was such a good review for me. This is one of those great areas in chess that gets short-changed with all my blitz playing

I'll message RT and see if he would be kind enough to edit my solutions. If not, it should read: "6. Kf6 Kh4 7. Kf5 Kg3 8. Kg5 +-

Not sure if it's exactly the same book, but it probably has similar content.

This came up in a game today, and I thought this was particularly instructive. The position went like this:


with white to move.

The corresponding squares here were d4 and d6. I remembered one of swingdoc's hints from the other problem: can we get this same position, but with black to move?

The winning line went: 1. Kd3 Kf5 2. Ke3 Ke6 3. Ke4 Kd6 4. f5

with white to win easily

sadly I think 1...Kd7!? stops the triangulation (Ke3 Ke7) and you have to repeat moves. 1. f5! gf 2. Kf4 creates a distant passed pawn and wins even if Black doesn't blunder.