ChessTriangulation is using your king to spend one extra move so that the turn passes to your opponent. In pawn endgames this situation comes up a lot: in the same position, if your opponent is to move he loses, but if you are to move you can't win directly. Then you walk your king around three neighboring squares, your opponent can't follow, and suddenly it's his move. It is the advanced version of the opposition, and once you know it you'll win a lot more pawn endgames when you play chess against the computer.
The key position
First, look at what happens if it were Black to move:
And with White to move? The king can't go forward: d6 and e6 touch the black king, and c6 is also covered by the b7 pawn. The only forward move is the pawn push c6+, and the tablebase says that's a draw (you'll see it under "Common mistakes").
So in the same position, Black to move loses, but White to move can't win directly. White's way out is triangulation: hand this exact position over to Black, unchanged.
How to win
The white king walks around a triangle
Why can't the black king keep up? It has only two squares that hold, d7 and c6: the b6 pawn covers c7 and the c5 pawn covers d6. With two squares it can only go back and forth; it can't make a triangle. When the white king steps back to d4, the black king goes to c6 to attack the c5 pawn; the white king defends it with Kc4, and now the black king's only move is Kd7. The white king returns to d5 and the triangle is complete.
If Black's first move is a retreat to the back rank instead (Kc8 or Kd8), the white king goes around the other way:
Every other Black move in these two lines loses even faster, according to the tablebase.
Break in and promote
The point is: don't rush the pawn. Walk around the triangle first and make Black move.
Common mistakes
The most impatient move is the immediate pawn push c6+, trying to open things up:
Another mistake is going around the triangle the wrong way:
When the black king is on c6, the white king must go to c4 (or stay on squares like d4 and d5 that guard c5). Never leave the c5 pawn alone.
Practice puzzles
Show answer
1. Kc4 Kc4! It defends the c5 pawn, the black king's only move is Kd7, and after Kd5 we're back in the key position with Black to move. According to the tablebase this is the only winning move: Kc3 and Kd3 draw, and Ke3, Ke4 or Ke5 let the black king take the c5 pawn, and then White even loses.
Show answer
1. Kd4 Kd4! It defends the c5 pawn. Whether the black king goes back to d7 or to b5, White follows with Kd5: the triangle d5, e5, d4 is complete and it's Black to move. According to the tablebase this is the only winning move; everything else lets the black king take the c5 pawn, and White loses.
FAQ
How is triangulation different from the opposition?
The opposition is when the two kings face each other with one square between them, and whoever has to move loses ground. Triangulation is a way to change whose turn it is: when the position is bad for you because it's your move, your king walks around three squares to spend an extra move and hand the turn to your opponent.
Why can't the other side triangulate too?
Triangulation works because the enemy king has less room: the edge of the board or pawns are in the way, so it has no three neighboring squares to walk around. In the example the black king has only two squares that hold, d7 and c6 (the white pawns cover c7 and d6), so it can only go back and forth. That's the reason.
When should I think of triangulation?
When you notice "if it were my opponent's move now, he would lose", but it is your move, ask yourself: can my king take a little detour and spend an extra move?
Try it yourself
Triangulation, the opposition and zugzwang belong together, so go back to chess endgames and read them as a set. Then play free online chess against the computer, get to a pawn endgame, count the squares and look for a chance to take a detour.