Corresponding squares 1
Corresponding squares can be a mighty weapon, I suppose. Therefore I need to be able to see the squares without the need to calculate them. In order to be able to do that, I must first understand what we actually are talking about.
I get the impression that there are a lot of systems out there that treat a part of the system. Opposition, diagonally and distant, triangulation and zugzwang seem to be all part of the same family. It might well be that the Trébuchet and the Reti manoeuver are siblings of the same family too. I wouldn't be surprised at all that there is a relationship.
What are we talking about?
Key squares
The first element that must be recognized is that there are such things as key squares. A key square is related to a pawn. If your king conquers a key square, you can make further progress. No matter who has the move! If the key square belongs to an enemy pawn that progress means conquering the pawn. If it is a key square of your own pawn, you can push it. Pushing your pawn is pushing the key squares with it. Which means that your king is no longer at that very key square, and the process start all over again.
Corresponding square
Your opponents king stands on a square. If your king steps on a square that corresponds with that square, your opponent has to give up that square. Think of opposition. If your king steps into opposition, your opponent has to give up the opposition.
Shortest distance between two key squares
Sometimes your king can choose between two key squares which have some distance.
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| White to move |
7k/1p6/1P2p3/1P2P3/4P1p1/6P1/8/K7 w - - 0 1
The numbers show which squares correspond. If the white king stands on the red 1, black must stand on the blue 1. If the white king stands on the red 2, black must stand on the blue 2 et cetera.
It turns out that the expression "key square" is used rather loosely. Maybe invasion square is better.
Goal
We can already see what the goal is. White moves to the shortest chain between two key squares, and the same does black. Once black steps on the chain, for instance on blue 2, white must step on the corresponding number red 2. Now black must make an awkward decision. If he goes to blue 1, we go to red 3. If black goes to blue 3, we go to red 1. This way we can invade by force.
Extending the numbers of the chain
You can imagine the square d3 (in contact with red 2 and red 3) corresponds with g7 (in contact with blue 2 and blue 3). There are several methods to expand the corresponding squares over the rest of the board. The authors have tried to simplify this. Meaning: they made rules for it which have the expected exceptions. Which makes things awkward. As is usually the case when I work with things I don't quite understand.
But anyway. The black king stands on h8. Which corresponds with a2. So if the white king steps on a2, black must leave it.
I'm going to try to remove the awkwardness of the diverse methods.
To be continued ...(hence the number in the title)
Apparently it is a method for double invasion. One move in the direction of two targets. Just like the Reti manoeuver. The targets can be anything. A promotion square, a pawn, attacking, defensive.
ReplyDeletePrevention of invasion.
ReplyDeleteIMHO, "invasion square" is a more appropriate description than "key square." In the example position, White has multiple targets that he may be able to reach; Black has none because there is no invasion route into White's position. White's "plan" is to utilize two weaknesses in order to force the Black King into a position from which it cannot defend both weaknesses. The process requires using distant opposition, slowly advancing toward both entry paths while preventing Black from maintaining opposition.
ReplyDeleteEuwe and Hooper's method for determining distant opposition is helpful. Here’s the idea, from a description from a comment on 7 JAN 2016.
Another "solution" to just "seeing" the Opposition is found in Dr. Max Euwe and David Hooper's book A Guide To Chess Endings, pg. 5, Diagram 6A.
This is entirely a visual solution. I'm going to describe it in words, because I cannot embed a diagram in Blogger.
Place the following letters on the following squares:
"A" on a2
"B" on b2
"C" on a1
"D" on b1
Using this group of 4 squares as an example, duplicate the same lettering on all of the rest of the 4-square groups on the board. (There are 16 groups of 4 squares on the board. Yeah, I know: you already know that - but can you "SEE" it?!?)
Euwe's Rule:
If you can move to a square that has the same letter as the opposing King, you have the distant Opposition; otherwise, you don't!
Initially, it will take some "looking" in order to "see" the group around one King and the corresponding group around the other King. There are some "speed ups" available. If the Kings are on the same color (VISUAL!), whoever has to move loses the Opposition.
To what move of white does that lead?
DeleteBased solely on distant opposition, 1. Kb2. But, as is already well known, opposition does not operate independently of intervening pawns in a vacuum, which cause a "disturbance in the Force." Mined squares, triangulation, outflanking, shouldering and corresponding squares all have a part to play. Most importantly, opposition is a means to an end, not an end in itself.
DeleteFEN: 7k/1p6/1P2p3/1P2P3/4P1p1/6P1/8/K7 w - - 0 1
I’ve been out of town for the last few days, so no real opportunity to dig into this position very far. I think I have figured out why White should NOT take the distant opposition on the first move.
In the given position, White has an impenetrable position; Black does not. There are two invasion routes into the Black position, connected by four squares. Using the Euwe/Hooper method of determining distant opposition (on a bare board with just the two Kings), White has four sets of four squares within which to maneuver; Black only has three sets. Black cannot advance toward the White King via the f6-square. That means that he can only maintain the opposition by moving back to h8, which (in the long run) is not helpful for preventing White from eventually invading via one of the two invasion routes.
The initial move sequence recommended by Stockfish is 1. Ka2 Kg8 2. Ka3 Kf8 3. Kb2, a “key” square for advancing the White King toward the potential invasion squares, with Black unable to counter it by taking the opposition. To gain the opposition, Black would have to move to either h8 (impossible with the BKf8) or to f6 (impossible because White controls that square with WPe5. Even if Black was to somehow get his King to h8, gaining the opposition, eventually that would be insufficient to protect both invasion routes.
The usual “advice” for endgames applies: Make haste slowly. White has moved one step closer to the invasion routes, and Black cannot stop it.
I’ll guess where this is going, even if I can’t “SEE” all the moves. Eventually, White will maneuver in such a way that after he occupies the crucial d3 square (with the Black King already on the crucial g7 square), Black will have to move. Since he can only defend both invasion points from g7, he will have to allow White entry through one of the invasion routes.
I could be all wrong. I’ll know after studying more of the proposed moves.
Today my copy of Auerbach's book will arrive.
DeleteThis is how Claude hallucinates:
-Dvoretsky reviews are good
-Dvoretsky talks about correspondent squares
Hence Dvoretsky is the best source for correspondent squares.
All squares are mined for white, except for one. If white loses the opposition, he cannot get it back by force. Which squares are mined changes with every move.
Whenever black steps on a blue square, white must step on the red square with the same number.
So the opposition should be maintained to the red squares. How black approaches the blue squares, dictates how white must approach the red squares.
I base my information sofar solely on this video. Which in itself is based on the book Comprehensive Chess Endings of Averbakh. The chapter about corresponding squares is written by Zinar.
ReplyDeleteIn the video there is a distinction between 3 systems. The 8-square system, the quadratic system and the triangular system.
In the video at 00:00, 29:00 and 43:00
These systems are simplifications to protect us from complexity. Since I don't like to use 3 simplified systems based on rules that I don't really understand, I try to work my way up to the understanding and then to simplify it myself. Because only then I can make informed decisions.
I'm experimenting with the systems. They seem to have a lot in common. So I try to unify them.
I don't have quite the gist yet.