### Triangulation in Chess

"A chess game is divided into three stages: the first, when you hope you have the advantage, the second when you believe you have an advantage, and the third... when you know you're going to lose!"
- Savielly Tartakower (1887-1956)

#### Definition

Triangulation, in chess endgames, is the art of giving up a tempo with your king in order to return to the same position with your opponent on the move. The idea being to put them in zugzwang and thereby achieve a favorable position or an all-out win for yourself. The term originates from the triangular moves that the king makes in order to relinquish a tempo.

#### Annotated Example

In Diagram 1 we see an illustrative example of king triangulation. Here white is wishing he could give up a move so that any move the black king makes loses to Kb6. White is in luck, since he or she can give up a move, so to speak, regain the position with black to move, and win the game!

```                       Diagram 1: King Triangulation
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 8
+---+---+---+---+---+---+---+---+
|   |   |BK |   |   |   |   |   | 7
+---+---+---+---+---+---+---+---+
|BP |   |WP |   |   |   |   |   | 6
+---+---+---+---+---+---+---+---+
|WP |   |WK |   |   |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
A    B   C   D   E   F   G   H
White to Move
```

1. Kd5 Kc8

Here white hangs on to his c pawn, while forcing black to the back rank. Black moves Kc8 to guard the queening square and maintain the opposition, as Kd8 or Kb8 lose very quickly to 2. Kd6 Kc8 3. c7 Kb7 4. Kd7.

2. Kd4

The key move in the triangulation. White steps back and in order to throw away a tempo.

2. ... Kd8

Black's king sees white's triangulation attempt and doesn't want to give him the benefit of stepping into the c7 square.

3. Kc4 Kc8

Now black is in a very unpleasant position. He's guarding the queening square in this move, but he can't stay there on his next move, so there's not much more he can do to defend.

4. Kd5!

We're now back in the position of move 1, but without black having access to the move Kc8 to save him. The same moves described above still lose fast, so black opts for a slightly slower loss.

4. ... Kc7 5. Kc5!

Now we are back in the position illustrated in Diagram 1 but with black on the move. The game ends quickly and there's nothing black can do.

5. Kc8 6. Kb6 Kb8 7. Kxa6 Kc7 8. Kb5 Kc8 9. a6 Kb8 10. Kc5 Kc7 11. a7 1-0