# Chess: Hartston's number explained

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EULER'S theorem, linking the numbers of sides, edges and vertices of a solid figure, states that E = V+S-2. So the number of edges of a cube, for example, is equal to 8 (its vertices or corners) plus 6 (its faces) minus 2, which comes to 12, which is the right answer. Bravo]

The little-known chessboard equivalent of this theorem is P+F = D+8, where P = number of your pawns, F = number of your open files, D = number of doubled pawns (with trebled and quadrupled pawns counting as 2, 3 etc) and 8 is the fundamental constant of file-preservation, known as Hartston's number after its discoverer.

A consequence of this theorem is that whenever you sacrifice a pawn, you gain an open-file in compensation. Most gambit openings then follow as simple corollaries of the theorem, with the Benko Gambit, in which Black jettisons his b-pawn for pressure along the a-file and b- file, as a prime example.

The game that knocked Gata Kamsky out of the Interpolis tournament in Tilburg was another that showed how an open file can be more important than the pawn that once stood on it. With 13 . . . Ng5] Artur Yusupov played a pawn sacrifice of great depth. Had Kamsky exchanged queens with 16. Qxd8 Raxd8 before taking the pawn, then 17. Nxe5 Bf6 followed by Nc4 would have given Black good pressure against the Q-side. Nevertheless, this would have been better for White than the way the game went.

By losing a pawn, Black gained an open e-file, but it was the bonus of the h-file, created by doubling his g-pawns, that proved decisive. Kamsky's 20. Nb3? was a bad mistake, brilliantly exposed by Yusupov's surprising exchange of bishop for knight. In the resulting position, Black's black squares were strong than White's white ones, with bishop on c5, queen on f4 and knight on e5 combining with the pawn on g4 to create a total grip on the position.

After 25 . . . g6] White was curiously helpless against the threat of Kg7 and Rh8. The finish came quickly, with 28 . . . Rh8 neatly ignoring the attack on his queen to threaten mate on h2, 29 . . . Qh6 renewing the threat, and 30 . . . Nxh4 completing the demolition. After 31. gxh4 Qxh4+ 32. Kg2 Qh3+ it is mate next move.

----------------------------------------------------------------- White: Yusupov Black: Kamsky ----------------------------------------------------------------- 1 e4 e5 17 Nf3 Rad8 2 Bc4 Nf6 18 Qc1 g4 3 d3 c6 19 Nd4 Bc5 4 Nf3 Be7 20 Nb3 Bxb3 5 0-0 d6 21 Bxb3 Nd7 6 c3 0-0 22 Ne3 Qf4 7 Re1 Nbd7 23 Qc2 Ne5 8 Nbd2 Nb6 24 Rad1 Rxd1 9 Bb3 Be6 25 Qxd1 g6 10 Bc2 h6 26 Qe2 Kg7 11 a4 a5 27 g3 Nf3+ 12 Nf1 Nh7 28 Kh1 Rh8 13 d4 Ng5 29 Nf1 Qh6 14 Bxg5 hxg5 30 h4 Nxh4 15 dxe5 dxe5 White resigns 16 Nxe5 Qc7 -----------------------------------------------------------------