EndgamesPlay as white

Triangulation

Learn how to lose a tempo with your king — three king moves around a triangle hand the move to your opponent and put them in zugzwang.

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Introduction

Triangulation is a king walk that loses a tempo — it gives away one turn. The king makes three moves around a triangle of squares and comes back to where it started — but now it is the opponent's turn. If the opponent is in zugzwang (every move makes things worse), that handed-over move decides the game. Look at this position. If it were Black's move, Black would lose: the king on d7 would have to step away, and White would break through with the c-pawn or march in with the king. But it is White's move, and White has no useful waiting move: after c6+ bxc6+ and the king takes back on c6, Black's king stops the lone b-pawn and it is a draw. So White uses the triangle d5–d4–c4: three king moves that keep the c5-pawn guarded all the way and bring the same position back, with Black to move. Black's king cannot copy the trick: it has only two safe squares, d7 and c6, and two squares make no triangle.

Lesson Content

Try this position

1. ♔︎d4

You want this exact position with Black to move. Start a king walk that keeps your c-pawn guarded.

♔︎d4 — the first step of the triangle. From d4 the king still guards the c5-pawn, so Black's king cannot take it. The plan is d5–d4–c4–d5: three moves, and the same position appears again with Black to move.

TriangulationZugzwang

Alternative Moves

c6+ — Draws! After c6+ bxc6+ and the king takes back on c6, only the b-pawn is left, Black's king gets in front of it, and the game is a draw.

♔︎c4 — Also wins, but one move slower. The walk that starts with ♔︎d4 hands Black the move fastest.

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1. ..♔︎c6

Black's king steps up to c6 and attacks the c5-pawn. The pawn is safe, because White's king on d4 guards it — but White must keep guarding it with every move, or Black's king takes it.

Triangulation

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2. ♔︎c4

Black's king is eyeing your c-pawn — take the next step of your walk without letting it fall.

♔︎c4 — the second step. This is the only winning move: c4 is the one square that keeps the triangle going and still guards c5. Any other king move leaves the pawn to Black.

TriangulationZugzwang

Alternative Moves

♔︎e5 — Loses! ♔︎e5 no longer guards c5: Black takes it with ♔︎xc5, then wins the b6-pawn as well, and Black's b-pawn is free to queen.

♔︎d3 — Draws! ♔︎d3 no longer guards c5: Black takes it with ♔︎xc5, and White's winning chances are gone.

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2. ..♔︎d7

♔︎d7 is Black's only legal move: the c5-pawn guards d6 and the b6-pawn, the b6-pawn guards c7, and White's king guards c5, b5 and d5. Black's king can only go back and forth between c6 and d7 — it has no triangle of its own.

Triangulation

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3. ♔︎d5

Close the triangle so that Black has to move.

♔︎d5 — the triangle is closed. White's king is back on d5 and Black's king is back on d7: the starting position is on the board again, but now ♗︎lack has to move. White lost a tempo with three king moves, and Black is in zugzwang.

TriangulationZugzwang

Alternative Moves

c6+ — Draws! From c4 the king cannot take back on c6: Black's king takes the c-pawn, then the b6-pawn, and the win is gone. Finish the triangle first.

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3. ..♔︎e7

Black must move, and every king move gives something up. ♔︎e7 is the longest defence, but the king no longer guards c6 — and that lets White break through with the c-pawn.

Zugzwang

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4. c6

Black's king has stepped away from the pawns — break through.

c6 — the breakthrough. If Black takes with bxc6+, the white king takes back and the b-pawn runs to b8. If Black does not take, cxb7 follows and the pawn queens on b8.

Passed PawnZugzwang

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4. ..bxc6+

Black takes, with check. If Black ignored c6, White would play cxb7 and queen on b8.

Passed Pawn

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5. ♔︎xc6

Black just took on c6 — win that pawn back and free your b-pawn.

♔︎xc6 — White takes back. The b6-pawn now has a free road to b8, and Black's king on e7 is too far away to catch it.

Passed Pawn

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5. ..♔︎e6

Black's king cannot reach b8 in time. ♔︎e6 changes nothing.

Passed Pawn

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6. b7

Your king guards the pawn's next square — keep the pawn moving.

b7 — one square from queening. The king on c6 guards b7, and nothing can stop the pawn.

Passed Pawn

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6. ..♔︎e5

Black cannot stop b8=♕︎.

Passed Pawn

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7. b8=♕︎+

Make a queen!

b8=♕︎+ — promotion, with check. King and Queen against King is an easy win. It was decided by the triangle ♔︎d4, ♔︎c4, ♔︎d5: three king moves that handed the move to Black and put Black in zugzwang.

TriangulationPassed Pawn

Key Takeaways

Triangulation loses a tempo: three king moves around a triangle bring back the same position with the opponent to move
It works when your king has a triangle of squares and the defending king has only two safe squares to wait on
Keep your weak points guarded on every step of the walk — here each square of the triangle protected the c5-pawn
Use it when the position is a zugzwang for your opponent but it is your move — first hand over the move, then break through

Now play it through

Step through every move on a real board, try the wrong ones, and ask Knighty why they fail.

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Related Lessons

The Trebuchet

Queen vs Pawn on 7th Rank

Rook Pawn Draw

Summary

You've learned triangulation. White wanted the starting position with Black to move, so the king walked the triangle d5–d4–c4–d5 — guarding the c5-pawn all the way — while Black's king had only two safe squares, d7 and c6, to wait on. After ♔︎d5 Black was in zugzwang, and the c6 breakthrough let the b-pawn run to b8.

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Vancura Position

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