Triangulation
The technique for losing a tempo with the king — taking three moves to reach a square the opponent reaches in two, and using the spare move to force the opponent into zugzwang.
Triangulation is the technique by which one side gives himself a free move — a “tempo” — in a position where the rules of chess would normally force him to move forward when he wants to wait. The king walks the long way around to reach a square he could have reached in two moves, taking three moves instead, and on arrival it is the opponent’s turn to move into a losing position. The procedure looks like a waste of two king moves; it is, in fact, the most economical use of three moves the position permits.
Why losing a tempo matters
The chess rules require that each player move on his turn; he cannot pass. In most positions this is no hardship. In endgames where one side is winning by opposition or by zugzwang, the requirement to move can be fatal — the side on move is the side losing, and what the winner needs is to find a sequence that ends with the loser on move.
Triangulation is the workhorse solution. The king has, in most positions, several different routes to the same destination. The two-square route takes two moves. The three-square detour — out, around, back to the same line — takes three. The difference is one move, and that one move is precisely what the player needs to flip the zugzwang from himself to his opponent.
The triangle
In the diagrammed position, the kings face each other on the e-file with one square between them. By opposition logic, the side on move — White — must yield. If White yields freely (Kd4, Kf4, Kd3, Kf3), Black’s king walks past and the position is lost.
The fix is to ensure that when the kings face each other again on the e-file, it is Black on move rather than White. White’s procedure: 1.Kd4 Kf6 2.Kd5. Now if Black plays 2…Ke7, White triangulates with 3.Ke4 — taking three moves (d4, d5, e4) to reach a square he could have reached in two — and Black, on move, must yield. The opposition has flipped.
Black is now on move and in direct opposition. Whichever way he steps, the white king advances to take a key square. The pawn on e5 — let us suppose it is Black’s — is now overworked, and White will win it by attacking the e5 square with both king tempo and pieces.
The triangle itself does not have to be on those particular squares. The principle is: find two king squares that are equidistant from the target, but require different numbers of moves to reach. The longer route is the triangulation route, and walking it loses you the move you wanted to lose.
Triangulation in practice
The technique appears most often in king-and-pawn endgames where one side has a pawn duo or a backward pawn and the other side’s king is trying to reach a defensive square. The defender wants opposition; the attacker wants the move passed to the defender. Triangulation answers the question of how to do that without giving up ground.
It also appears in bishop endings — particularly in endings of the same-coloured bishops, where one side’s king can wait while the bishop covers two duties. The technique is identical: the king traces a three-square triangle while the bishop stays static, and the opponent’s bishop is eventually overworked.
A classical example: Capablanca v. Tartakower, New York 1924, the king-and-pawn ending after move 35. Capablanca’s king walked from f4 to e5 to e4 to f4 — a triangle that gave Tartakower the move in a position where Tartakower’s king had only one legal move and that move lost. The game was decided not by calculation but by counting.
The lesson is geometric and almost combinatorial. In any king-and-pawn ending, the number of king moves you can spend is fixed; the number of king moves you must spend before the position resolves is determined by the geometry of the pawns. Triangulation is the technique for ensuring that the moves you must spend land on the right side of the parity. Master it and you will routinely win positions that lesser players draw — and draw positions that lesser players lose.