Kasparov sacrificed a pawn for piece play and it worked- Alburt had to play very carefully to avoid a disaster- but even making it to a Queen ending didn't help him, with Kasparov's accurate zwischenzug 37...Qb1+!, forcing the Queens off the board and preventing 38.Qd8+.

31...Qa3 32.h3 a5 33.Kh2 Qb4

55...Ke5 56.Ke1 Kf5 57.Kf1 Ke4 58.Ke1 Ke3 wins.

55...Ke5 56.Kg1 Kf5 57.Kf1 Ke4 transposes.

<55...Kf5! 56. Kg1>

[56. Kf2 Kf4 57. Ke1 (57. Kg1 Ke3 58. Kf1 f2 59. Kg2 Ke2 60. Kh2! Ke1! 61. any f1=Q+! ) 57...Kg3 (57...Ke3 also wins for Black) 58. Kf1 f2 59. Ke2 Kg2 ]

<56...Ke5! 57. Kf1>

(57. Kf2 Kf4 ; 57. Kh2 Kf4 58. Kh1 Ke3 59. Kg1 Ke2 ; 57. Kh1 Ke4 58. Kg1 Ke3 59. Kf1 f2 60. Kg2 Ke2 61. Kh2 Ke1! )

<57...Ke4> 0-1

Black has won the position via triangulation as now it is the exact same position as the start of the puzzle only with White to move. This is bad for White as he is in zugzwang and must surrender the opposition to Black.

For example,

A) 58. Kf2 Kf4

B) 58. Ke1 Kg3 (58...Ke3 also wins for Black) 59. Kf1 f2 60. Ke2 Kg2

C) 58. Kg1 Ke3 59. Kf1 f2 60. Kg2 Ke2 61. Kh2! Ke1!

55.........Ke5
56.Ke1 Kf5
57.Kf1 Ke4
By 21/2 moves, Black arrived at exactly the same puzzle position. The only difference now is that it is White's move:

With the paucity of pawns, this position offers a fairly simple example of coordinate square theory: Each king seeks to move to a given square in response to the other reaching a corresponding square, leaving the other in zugzwang. Most of us are familiar with this idea in its simplest form: the opposition, when whoever gives ground first must abandon hope.

Here Black wins by triangulation, moving so as to return to the initial position — this time with White to move:

55. ...♔f5!

Note: ...♔e5! also wins by transposition.

56. ♔e1/♔g1

If 56. ♔f2?, ♔f4 .

56. ...♔e5;
57. ♔f1, ♔e4!

Now we see the point of Black's maneuver. White must move, and has three choices:

(1) 58. ♔f2, ♔f4 (the opposition);
59. ♔f1, ♔g3;
60. ♔g1, ♔xh3

With two pawns, Black can easily find a tempo to dislodge the white king from a queening square.

(2) 58. ♔e1, ♔e3 (the opposition);
59. ♔f1; f2!;
60. ♔g2 ▢, ♔e2

White can't stop the pawn queening.

(3) 58. ♔g1, ♔e3 and White can pick his poison:

(3.1) 59. ♔f1, f2;
60. ♔g2 ▢, ♔e2 as in variation (2).

(3.2) 59. ♔h2/♔h1, f2;
60. ♔g2, ♔e2;
61. ♔h2 (White's last hurrah, hoping for stalemate), ♔e1!

Without laying any claim to the shortest line, I think this is what would happen.

55...Kf5 56. Kg1 Ke5 57. kg1 Ke3 58. Kf1 f2 59. Kg2▢ Ke2

a) 60. Kh1 f1=Q+ 61. Kh2 Qf4+ 62. Kg2 Qg5+ 63. Kh1 Kf3 64. Kh1 Qg2#

The first key position is this one:

click for larger view

If we arrive at this position, the outcome of the game will depend on who is to move. And it turns out that you want the other guy to be the one about to move.

If it's white to move, his king has to give way and the black pawn crashes through ... Ke1/f2 or Kg1/f2. The pawn arrives safely on the seventh rank without giving check, which usually means game over.

If it's black to move, he can't make progress. He doesn't want to play 1...f2+ because 2. Kf1 either steals the pawn or forces 2...Kf3 stalemate.

By the same logic, another key position is this one:

click for larger view

If it's black to move, his king has to retreat. That means he is no closer to queening a prawn.

On the other hand, if it's white to play, he has to retreat. This allows black to push on with Ke3 or Kg3 and then the f-prawn will queen.

So rewind a move each and we get to our puzzle position:

click for larger view

The first thing to note is that the immediate 55...Ke3 doesn't work. After 56. Ke1 we are into our first key position, but with black to move. And as we've seen, that doesn't work.

Nor can black play 55...Kf4 straight away. Then white replies 56. Kf2 and we are into our second key position and again its black to move.

This means that our puzzle is another of those positions where you don't want to be on the move. But it only works for white if we land on exactly this position with black to move. In other words, if the black king is on e4 (threatening Ke3), the white king must be on f1 (to answer Ke3 with Ke1).

The trick here is that white has a lot less space than black. His king is on the back rank, which means that he only has one rank to work with. By contrast, the black king has several ranks to work with.

This means that white can get to and from f1 in exactly two moves (Kf1/Ke1/Kf1) or (Kf1/Kg1/Kf1), but he can't get from f1 and back again in three moves. By contrast, black can get to e4 in either two or three moves. The three move quickstep is a little triangle of (Ke4-Kf5-Ke5-Ke4) or the equally good (Ke4-Ke5-Kf5-Ke4).

In other words, by taking three moves to white's two, black can recreate the puzzle position, but with white to move instead of black. And that means that he can force either of the two key positions with white to move.

Aha! I hear you cry. White does have two ranks to play with. He can triangulate himself by playing Kf2 at some point. So what if we get something like... 55...Kf5 56. Kf2

click for larger view

That does indeed cut across black's fancy stepping on f5 and e5. And if 56...Ke4 57. Kf1 repeats the position. But black simply reverts to key position two with 56...Kf4 and wins.

If it was whites move black could easily march straight over & capture the white pawn or shepered his own pawn home. Black can create that situation by moving his K in a small 3 move triangle e5>f5>e4 or f5>e5>e4, white is now in Zugzwang, any move loses.

(if after moving to the 5th rank white plays Kf2, then Kf4 wins even quicker)