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Guided lesson · Intermediate · Endgames

The geometry of the king: two objectives at once

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There's an intuition we bring from life that chess breaks: we assume going in a straight line is faster than going diagonally. On a board it isn't. The king takes just as long to travel seven squares sideways as seven diagonally, and that equality allows something that looks impossible: walking towards two places at once. In 1921 Richard Réti composed a study on this idea that is still the best geometry lesson ever given — a white king trapped in a corner, a black pawn racing to queen and a white pawn nobody is supporting. It looks lost on both counts. It's a draw, and in this lesson you're going to play it yourself.

The key idea

The king moves diagonally at the same cost as in a straight line, so a well-chosen diagonal chases two objectives at once: closing in on the enemy pawn AND escorting your own. When an endgame looks lost on two fronts, look for the square from which you threaten both.

Example 1 · Réti's study

White king on h8, white pawn on c6, black king on a6 and black pawn on h5. Count the squares: the white king arrives late to everything. And it draws.

  1. Kg7 Before moving, do the count everybody does: the black h5 pawn queens in four moves and my king is six squares away from it. And my c6 pawn doesn't queen either, because the black king is right beside it. It looks lost twice over. The answer is this move — the king does NOT run at the h5 pawn: it walks diagonally, towards the centre.
  2. h4 The black pawn runs, as it should. It still looks like it gets there on its own.
  3. Kf6 And here's the magic, which is pure geometry: from f6 my king does TWO things at once. It keeps closing in on the h4 pawn along the diagonal, and at the same time it enters the square of my own c6 pawn, ready to support it. One move, two threats — the king hasn't gone to one place, it has gone to both.
  4. Kb6 The black king has to deal with my pawn, because if he leaves it, it runs and queens. And the moment he deals with that, he stops watching the other half of the board.
  5. Ke5 The diagonal again, two objectives again. My king closes in on the black pawn and supports mine in the same move. This is the study's entire mechanism, and it fits in one sentence: diagonally, no time is lost.
  6. h3 The pawn keeps running and no longer gets there alone: my king is inside its square. Drawn. Keep the feel of this position rather than the moves — when you count squares and find you're late to both places, check whether there's a diagonal that serves for both. Very often there is.

Example 2 · The same trick, another position

White pawn on c2, black pawn on h5 and the kings crossed. The same idea, so you can see it wasn't a one-off.

  1. c4 Here my pawn can run, because his king is far away. I push, and along the way I force the black king to decide: either chase my pawn or push his own.
  2. h4 He chooses to run. Both pawns at once, and the winner is whoever arrives better escorted.
  3. c5 I keep going. Count the tempi: both pawns queen at almost the same time, and when that happens the endgame is decided by what happens AFTER queening, not by who gets there first.
  4. Kf4 The black king comes closer to escort his pawn, which is correct on his part.
  5. c6 And I keep running. My king on b6 is doing the same double job as in the previous example: it escorts my pawn and it's close enough to the centre to get in the way of his if it had to.
  6. h3 Drawn again, by the same mechanism. And that's why this lesson exists: the geometry of the king isn't a pretty study from 1921, it's a tool that turns up every time two pawns are running on opposite sides. When it happens to you, don't count in straight lines.

Practice

Réti's study, and now it's your turn. His pawn queens in four and your king is six squares away. Don't chase it: look for the diagonal that serves for both jobs.

  1. Diagonally, not in a straight line. Chasing the pawn along the eighth rank would arrive late; this doesn't. Don't chase the pawn head-on. Move the king towards the centre, diagonally.
  2. There's the double threat: you keep closing in on his pawn AND you enter the square of your own. One move, two objectives. Continue along the same diagonal, to a square that serves both to reach his pawn and to support yours.
  3. And both things again. His king will have to deal with your pawn, and then his own is left without an escort. Keep coming down the diagonal, without stopping threatening both things.

You've just saved an endgame that looked lost on two separate fronts, and without a single complicated move. The underlying idea works for any pawn endgame: on a board, the diagonal costs the same as the straight line, so a king walking diagonally can chase two objectives at once. When you count squares and find you don't get there, count again — but diagonally.