Triangulation: spending three moves on a two-move journey

You have exactly the position you want and it is the wrong person's turn. The king walks a little triangle and hands the problem back.

The cost Three king moves for a two-move journey
Needed when Every pawn is frozen and you must pass

Triangulation is losing a move on purpose. Your king walks to a square by the long way round, three moves instead of two, and the position that comes back is identical except that your opponent is the one who has to move. It exists for one specific situation: you have the position you want, and the wrong side to move.

It only comes up when you have no pawn move left to waste. If a pawn can take a step without spoiling anything, do that instead, because it is simpler and it cannot go wrong. When every pawn is frozen, the king has to do the waiting.

What White wants: this position, with Black to move

Both pawns are frozen; nobody has a pawn move. Black to move, and the king has four squares. Not one of them still guards e6, so White takes the pawn next move and ends up with a king in front of his own pawn, which is the winning shape from opposition.

The triangle

The same position with White to move, which achieves nothing at all. There is no pawn to push, so the king goes for a walk: d7, c7, c8, back to d7.

1.Kc7 Ke7 2.Kc8 Kf7 3.Kd7

Three moves out and back for White, two for Black, and the identical position has returned with Black to move. The extra square c8 is the one the black king has no way of matching.

The reason it works here is that the two kings do not have the same freedom. The black king is tied to the small set of squares that keep e6 defended, and White is not tied to anything. Whenever one king has a spare square that the other cannot copy, the tempo can be handed over.

Be honest about the limits, though. Black has other tries in this position, and working out whether each of them holds is a corresponding squares problem, which is the systematic version of this counting and a great deal harder than one triangle. What is not in doubt is the mechanism: three moves for a two-move journey changes whose turn it is, and nothing else on the board changes at all.

Look for it when the position is right and the move is wrong.
It needs a spare square your opponent's king cannot mirror. If both kings have the same freedom, nobody gains anything.
The victim ends up in zugzwang, forced to make the move that loses.
It is not only a king trick. Queens and rooks triangulate too, wherever the same square can be reached two different ways.
The rest of the family lives in pawn endgames.

Questions people ask

What is the difference between triangulation and opposition?

Opposition is a description of a position: who is standing where, and who is therefore going to be forced to give way. Triangulation is a method for changing whose turn it is, so it is the tool you reach for when the opposition is currently against you. You triangulate in order to get the opposition, which means the two ideas are almost always used in the same move.

When does triangulation not work?

When your opponent has the same amount of room as you do. If the defending king can walk its own triangle in reply, the tempo comes straight back and you have achieved nothing but a repetition. It also fails when your king cannot leave its square without giving something up, which is why the manoeuvre needs a square that is genuinely spare rather than merely empty.

Triangulation sits under Endgame words in the reference. Nearby: