Corresponding Squares and Triangulation

Here is a classic example of Corresponding Squares and Triangulation:

Pairs of Reciprocal Zuzwang exist:

d6(w) and d8(b)
c5(w) and c7(b)

The black zugzwang squares, d8 and c7, border c8. The white zugzwang squares, d6 and c5, border d5. Therefore, d5(w) and c8(b) are also corresponding squares in this position. This essentially means:

• If White moves his King to c5, Black must be able to play Kc7


• If White moves his King to d6, Black must be able to play Kd8


• If White moves his King to d5, Black must be able to play Kc8

The sets of squares are said to be in correspondence.

Black must avoid moving into a corresponding square without white already occupying the other square in correspondence:

• If Black plays Kc7, White can win with Kc5


• If Black plays Kc8, White can win with Kd5


• If Black plays Kd8, White can win with Kd6
  
  So, the goal for white is to force Black to move in the current position, because c8(b) and d5(w) are in correspondence. Unfortunately, it is White's move.
  
  White must lose a move with Triangulation.
  
  With the Black King on d8, white will move from c4 to d4 (or reverse) to force black onto c7 or c8. White only then occupies the corresponding square which black moves to:

1.Kc4 [ 1.Kd6 Kd8 d6/d8 correspond and white cannot make progress with this move.; 1.Kc5 Kc7 c5/c7 correspond and white cannot make progress with this move.]

1...Kd8 Kd6 would now win for white, as d6/d8 correspond, but he cannot play that...

2.Kd4 distant opposition as well as zugzwang. Now Black is forced onto c7 or c8 and White wins by occupying the corresponding square.

2...Kc8 3.Kd5! because d5 and c8 correspond.

3...Kd8 [ 3...Kc7 4.Kc5(.) followed by Kb6+-]

4.Kd6 because d6 and d8 correspond.

4...Kc8 5.c7 and white wins.

Go over this fine example several times until you fully understand it perfectly. It is one of the most important endgame positions to know. Below is another example of Corresponding Squares. See if you can identify the corresponding squares in the same method as above. The key is to identify mutual zugzwang pairs of squares and identfy common squares between them that lead to additional corresponding squares.

Highlight below to see answer
[White needs to break through at e3. The obvious corresponding squares are

d2(w) and f3(b)
c3(w) and e3(b)

Looking at adjoining squares:

d2(w) and c3(w) adjoin c2(w)
f3(b) and e3(b) adjoin f4(b)

Therefore:
c2(w) and f4(b) correspond.

But white has the similar additional resource of triangulation using b2 and b3 squares, as both those squares correspond to the single square f3 for black, and that is key to the win, allowing White to triangulate at an opportune time to place black in zugzwang.
]