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Triangulation

Intermediate Must Know

Triangulation is the art of transferring the move to your opponent. You reach a position that would be winning if only it were their turn — so you "lose a tempo," either by walking your king in a triangle or by spending a spare pawn move. With no such waiting move of their own, the opponent falls into zugzwang.

The Concept​

In many king-and-pawn endings the player to move is worse off — moving their king means giving way. If you can hand that obligation to your opponent, you win. Triangulation does exactly that.

Winning by Outflanking​

FEN: 8/3k4/8/3Pp3/4P3/3K4/8/8 w - - 0 1

White has a protected passed pawn on d5 (guarded by e4), but Black's king blockades from d7. Charging straight ahead achieves nothing — so White's king sidesteps to lose a tempo and outflank:

1.Kc4! Kd6 2.Kb5 Kd7 3.Kc5 — White's king has worked its way to c5. Black cannot keep guarding the d-pawn and meet the king's invasion. White wins (Stockfish: decisive).

A Spare Tempo​

FEN: 8/3k4/4p3/3KP3/8/P7/8/8 w - - 0 1

Here the e-pawns are locked and the kings mirror each other — king moves alone get nowhere. But the a-pawn is a reserve tempo:

1.Kc5 Kc7 2.a4! — passing the move. Now Black must give way: 2...Kb7 3.Kd6 Kb6 4.Kxe6 and White's e-pawn marches home. The spare pawn move did the same job a king triangle would.

When Triangulation Fails​

FEN: 8/2k5/8/8/2K5/8/8/8 w - - 0 1

With no pawns and equal king mobility, neither side can ever be put in zugzwang — whatever White tries, Black mirrors it. Draw. Triangulation only wins when your king has a waiting move and the opponent's does not.

Exercises​

Exercise 1​

FEN: 8/8/2k5/8/2K5/8/2P5/8 w - - 0 1

White to move. How does White win?

Solution

White's king is already ahead of the pawn, so it escorts it through:

1.c3 Kd6 2.Kb5 Kd5 3.c4+ — the pawn rolls forward under the king's protection and queens. White wins.

Exercise 2​

FEN: 8/8/4k3/4p3/4P3/4K3/8/8 w - - 0 1

Can White win with triangulation here?

Solution

No — draw. Black has exactly the same squares as White (d6/e6/f6 mirror d3/e3/f3). Whatever triangle White walks, Black walks the matching one and never runs out of waiting moves. The position is a dead draw.

Summary​

  1. Triangulation transfers the move to your opponent
  2. The tool: lose a tempo — a king triangle, or a spare pawn move
  3. The result: the opponent is in zugzwang and must give way
  4. Requirement: you have a waiting move and they don't
  5. Counter: if the opponent has equal waiting moves, it's a draw