Friday, June 23, 2017

Endgame Strategy: Corresponding Squares

Two of the most important aspects of chess are recognition of geometric patterns and calculation. These two concepts alone lead to many other aspects of chess. One of those is a term you've probably heard thrown around at times at the club called "Opposition", and many times in king and pawn endgames, who has the opposition very often decides who wins or if the game in a draw. Sometimes it doesn't matter and one side is winning no matter what, or the position may be drawn in both cases. But often times, it means the difference between a position being won for one side versus the other side being able to draw the position.

The subject of this article, Corresponding Squares, is an extension of the concept of opposition, and is often necessary to determine the result of a position, particularly those in king and pawn endgame positions where no more than one pawn is mobile. The concept of corresponding squares is also where other concepts are derived from, particularly triangulation, which you will see in this article.

I am going to start with a very basic position where the knowledge of opposition is sufficient to figure out the result, but I am going to expand the thought process to corresponding squares. I will then go through two other well known positions that are more complex that require corresponding squares to figure out. I will then provide three positions for you to figure out yourself followed by the solution to all three.

Let's start with the following basic position:

There is a three step process to determine whether White can win or if this position is a draw.

Step 1 - Determine the winning zone for White.
Step 2 - Determine all of the corresponding squares that are relavant.
Step 3 - Determine whether or not White can win.

Let's start with the winning zone. Those familiar with king and pawn versus king positions will know that White's winning zone is two squares in front of the pawn along with the square to its left and right. So in this case, that would be the squares c5, d5, and e5. If the White King can get to any of these squares without moving his pawn, he wins.

Now we need to figure out the corresponding squares. Let's start with d4. If the White King goes to d4, he is adjacent to all three winning zone squares. There is only one square that Black can get to that covers all three of those squares and that is d6. Therefore, we will label d4 and d6 with a 1. Next, the square c4 is adjacent to the winning zone squares c5 and d5 along with a "1" square, namely d4. The only square on Black's side of the winning that is adjacent to c5, d5, and his "1" square is c6, and so c4 and c6 both get a 2. The square e4 is adjacent to winning zone squares d5 and e5 along with his "1" square. The only square Black has adjacent to all three of those is e6, and so e4 and e6 will be marked with a 3. The square b4 is only adjacent to c5 and his "2" square. There is nothing new to contribute to this square as if White ever goes to b5, Black can answer with d5 and the White pawn will fall. Therefore, since nothing is new and we only need to be adjacent to c5 and 2, the "1" square is also adjacent to those, and so b4 will also get a 1. The same can be said about f4 in that there is nothing new and is only adjacent to e5 and "3", and going to f5 again allows the Black King to d5 and the pawn can't be saved, and so f4 also gets a 1.

Therefore, below is the diagram with the winning zone marked by X's and the corresponding squares labelled.

And so now we reach Step 3. Is there any way for White to get to a square that Black can't reach? Since there is left-right symmetry and the edge of the board is not a factor, it doesn't really really matter whether he goes to c3 or e3, and staying on the second rank makes no progress for White. So let's assume White goes 1.Ke3. He is adjacent to a 1 square and a 3 square, and so Black cannot go to c7, but whether he goes to d7 or e7 doesn't matter. So let's play 1...Ke7. After 1.Ke3 Ke7, if White goes to d4, which is a "1" square, then Black must go to the corresponding square, namely d6, and if White goes to e4, which is a "3" square, then Black must go to his "3" square, namely e6. If we continue to make moves for White and the corresponding moves for Black, we will quickly recognize that there is no sequence of moves for White that lead to occupying a square with a number that Black can't reach, and so therefore, this position is a draw provided that Black continues to follow the corresponding squares, or grab the pawn if White completely abandons it.

The next example is a more complicated case.

So here we again first determine the winning zone for White. In this case, it's in two separate areas of the board. If the White King can get to a3, b3, e2, or f2, he wins. Therefore, the first question is how quickly get each get from one side to the other? The square a2 is the only square for White adjacent to the winning zone as b2 is attacked by the Black pawn. To get to the winning zone on the right side requires occupancy of the e2 or f2 squares by the White King. So the quickest way to get there is a2 (1) to b1 (2) to c1 (3) to d1 (4). For Black, the shortest route is also 4 squares, namely b4 (1) to c5 (2) to d4 (3) to e3 (4). Next, the square e1 gets a 5 as now f2 can also be reached from White's current location, and he can still go to e2, and so Black needs a square next to the "4" and both winning zone squares, and so that makes the f3 square a 5. There is one place that White can try to triangulate and that is on a1. It is adjacent to the 1 and the 2. So a1 gets a 6 and for Black, the corresponding square that covers all of those on the Black side is b5.

Therefore, we have the following diagram:

Black, therefore, must go to the square labeled with a 5, namely 1...Kf3. There is nowhere for White to triangulate such that Black can't do the same thing. Once again, we have a drawn position.

The last example before the exercises I am going to show you a case where White wins.

So again we start with the winning zone. The winning squares for White are those which he can shoulder the King out or win the b-pawn. These are d4, e3, and e2. Therefore, if you label the square the White King occupies as a 1, it is next to both e2 and e3 from the winning zone. Now keep in mind that the White pawn on d3 being a passer prevents Black from going below the 3rd rank. Therefore, the only square for Black is f3, which he currently occupies, and so f3 gets a 1. Next you have c4. It is adjacent to the d4 winning square and the 1. The only square that the Black King can occupy that is adjacent to both d4 and the "1" square id e3, and so e3 gets a 2. Next is c2. It is adjacent to the 1 and 2 squares, and so it is labelled a 3. The only legal square for Black that is adjacent to both his 1 and his 2 squares is f4 as f2 would put the black king outside the box of the White passed pawn. Now we look at b2 and b3. There is nothing new about the squares that they are adjacent to. 2's and 3's are adjacent to both, and the 1 square is adjacent to both, and so both b2 and b3 get a 1.

So now you have the following:

And here we see our first example that leads to a win. White should move his King to c2, and Black must go to his corresponding square, f4, with the King. Next White should move his King to b2 or b3, and Black must go to his square labelled a 1, f3. Now White goes to the other square marked with a 1, and Black has no corresponding square to go to because e4 is covered by the White pawn on d3. Otherwise, this would be Black's other "1" square and we'd have a draw.

So thus far we have 1.Kc2 Kf4 2.Kb2 Kf3 3.Kb3 and now if 3...Ke3, a 2 square, White will go 4.Kc3 and White wins as it is Black to move and he must move away from d4 and White gets into the winning zone. If Black plays 3...Kf4, then White goes 4.Kc2 and again, Black has no square to go to. If he goes 4...Kf3, then 5.Kd2 wins as we have the starting position again, but this time with Black to move, and Black will have to move away and allow White into the winning zone. If instead Black goes 4...Ke3, then White goes 5.Kc3 and again we see Black in Zugzwang and White will get his King into the winning zone via d4.


Now I want to you look at the following three positions and use the concept of corresponding squares to try to figure out the correct result for each of the following three positions. In the first one it is Black to move. In the final two, it is White to move.

Think you got all three right? Let's look at them and find out.

The first problem sees the exact same position as the second example with everything moved up a square. The same winning zone and corresponding squares exist, again everything shifted up a square. Therefore, we have the following position.

There is one major difference here. Once again the white king resides on a 5, so Black must play 1...Kf4, but here, White has space to retreat backwards and is not up against the edge of the board, and so in this case, White wins with 2.Kd1! as he now threatens to occupy the 3, 4, or 5 square on the next turn, and there is no corresponding square for Black as the White pawn covers the e5 square. Therefore, if Black plays 2...Ke4, then White goes 3.Kd2 and there is no "4" square for Black to move onto and White will be able to enter the winning zone on one side of the board or the other, depending on how Black reacts. If Black plays 2...Kf5, then White wins with 3.Kc2 as Black is out of reach of the "3" square, and White will enter the winning zone over on the queenside.

The second problem sees Black being the side trying to win. Like the very first example, if he gets two squares in front of his passer, he wins, and so e3 and f3 are part of the winning zone. In addition, he also wins if he gets to e2 or e1 as he can then aid the f-pawn to promotion, or if White goes after the f-pawn, then Black can maneuver his king such that once White takes on f5, Black will go to f3 with his king, shouldering the white king out, and will keep white far enough away so that he can capture the pawn on h2 and get back out of the way of his own h3 pawn by one move and promote the h-pawn.

Therefore, we have the following diagram of corresponding squares, geared toward keeping the King off the e-file. Until Black tries to come in from the side, White will just toggle between e2 and f2 to keep the King out of e3 and f3.

Here, White draws with 1.Kf2. The only way for Black to attempt to make progress is with 1...Kd3 and now we use our corresponding squares on the f-file to keep the black king from ever entering the e-file. 2.Kf3 Kd2 3.Kf2 Kd1 4.Kf1 Kd2 5.Kf2 etc.

The final position takes a little more work to figure out the winning zone. There are two critical cases. If the white king ever occupies f4, the g-pawn will fall and white wins. The more difficult one to see is that if the white king gets to d4, he threatens the move e5 as after a trade on e5, white would threaten both Kd6 and Kf4 and the black king can't be in both places at one time, and so one of the pawns falls and white wins.

Therefore, when white goes to d4, black must prevent the pawn push by putting his king on f6, and when white goes to e3, black must stop Kf4 by going to g5. So we start by marking d4 and e3 along with f6 and g5 with 1 and 2. Next, d3 gets a 3 as it is adjacent to both 1 and 2 and g6 gets the same assignment for black. Next, we can go along the outside of these three squares for each player by labeling them four through 8. You will notice in both cases, the same numbers are adjacent to any given number on both sides. For example, on both sides, you can get from the 4 to the 1, 3, or 5. Since we have only two critical squares, a 3-by-3 grid is sufficient, and all squares beyond that will be the same number as that two squares away, working outward from the critical 3-by-3 block. So, for example, b4 is a 1 because it can get to 3, 4, or 5 just like the 1 can from d4. The same is done for Black on the 8th rank, mimicking the 6th rank.

So that leads to the following diagram:

We can see now that Black currently occupies a square marked with a 3, and if white now moves to a square with a 3, black has no 3 to go to, and White will win by working his way toward the winning zone going on squares that either match the number black occupies, or going to a number that black can't reach. So an example would be 1.Kb1 (the only move that wins) Kf7 (moves like 1...Kg7 are easier for white, he simply moves to the "5" square, namely 2.Kc1) 2.Kc2 (there is no way for Black to move all the way to the "6" on his side from f7) Kg6 3.Kd3 (in essence, White has the opposition now and therefore wins) and now if 3...Kg5, then 4.Ke3 and Black must move away and give White the f4-square for his King. If instead, 3...Kf6, then 4.Kd4 and again, black must now allow the e5-pawn push. If black moves away from the "1" or "2" square, like if he goes to f7 or h5, then white goes to either the "1" or the "2", depending on which one is out of Black's reach. For example, after 4...Kf7, white wins with 5.Ke3 and after 4...Kh5, white wins with 5.Kd4.

Well, that concludes this article. The next time you have a king and pawn endgame with predominantly static pawns, before you just assume that you are winning because you are up a pawn or lost because you are down a pawn, use the corresponding squares method to see who has the opposition from a distance. It can easily mean the difference between a win for one side versus a draw for the weaker side. Snatch half a point away from your opponent, especially if he had a win and goes to the wrong square where it is then a win no more!

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