A fresh approach

Reimagine democracy. Reimagine capitalism. Reimagine education. For all the reimagining thrown at big ideas, they don’t seem much perturbed. You can reimagine a problem too, but it probably won’t be fruitful. It won’t help you find the end of the Sellotape, or balance the books (unless you worked for Enron).
 
But some problems really are amenable to a fresh perspective. According to legend, the mathematician Carl Friedrich Gauss was tasked at primary school with adding up the numbers from 1 to 100. The obvious approach is (to quote the King of Hearts from Alice in Wonderland) ‘Begin at the beginning, and go on till you come to the end: then stop.’ Gauss saw the problem in a different light. He spotted that the numbers split into 50 neat pairs: 1 + 100, 2 + 99, 3 + 98, which each add up to 101. There are 50 such pairs, so the problem shrinks dramatically. Just multiply 101 by 50 to get the answer: 5050.
 
You have to admire Alexander Grischuk’s imaginative problem solving at the Fide Grand Prix this month in Hamburg. Playing White against Maxime Vachier-Lagrave, he reached the advantageous endgame shown in the diagram. The bishop plays well in an open position with pawns on both flanks. Black’s knight is yoked to White’s passed pawn on a4. All very well, but how would you win the game?
 
The most plausible approach is to shuffle the white king westwards to b4, to budge the knight. Plausible, but futile, as Black can impede this with his own king. If Black’s d5-pawn were enticed forward, it would easily be captured. But that nuance hardly looks relevant. The requisite bishop manoeuvre takes time, and Black can defend with his king.
 
Grischuk’s solution is striking. He sends his king in precisely the opposite direction, to h5, even though the h6-pawn is easily defensible. Next he attacks the d5 pawn, which must advance, as Black’s king and knight are shackled at either side of the board. With the pawn dislodged and vulnerable, Grischuk patiently optimises his bishop. Once all the pieces are in place, the king switches direction, decisively.
 
Grischuk–Vachier-Lagrave
Fide Grand Prix, Hamburg, November 2019
 
1 Nf3 Nf6 2 c4 c5 3 Nc3 d5 4 cxd5 Nxd5 5 e3 e6 6 Bc4 Be7 7 O-O O-O 8 Qe2 Nc6 9 Rd1 b6 10 d4 Nxc3 11 bxc3 Qc7 12 e4 Rd8 13 Be3 Na5 14 Bd3 Bb7 15 h4 b5 16 Bxb5 Bxe4 17 dxc5 Bxf3 18 Qxf3 Bxc5 19 Bf4 Qb7 20 Qe2 Qe7 21 h5 Bd6 22 Bxd6 Rxd6 23 Rxd6 Qxd6 24 Rd1 Qc5 25 Rd7 Qxc3 26 Qd1 h6 27 g3 Rf8 28 Rxa7 Nc6 29 Rd7 Ne5 30 Rd8 Qc7 31 Rxf8+ Kxf8 32 a4 Ke7 33 Be2 Qc3 34 Qb1 Nc6 35 Qb7+ Kf6 36 Bf1 Na5 37 Qd7 g5 38 hxg6 Kxg6 39 Kg2 Kg7 40 Bb5 Qe5 41 Qd3 Qc5 42 Qf3 Qd4 43 Be8 Qd5 44 Qxd5 exd5 (see diagram) 45 Kf3 Kf6 46 Kg4 46 Ke3 Ke7 47 Bb5 Kd6 48 Kd4 f6 49 Kc3 Kc5 leaves no way forward. 46… Ke7 47 Bb5 Kf6 48 Kh5 Kg7 49 Bf1 Nb3 50 Bg2 d4 51 Bf1 Nd2 52 Bd3 Nb3 53 Bf1 Nd2 54 Bd3 Nb3 55 Bb5 f6 56 Bf1 Na5 57 Ba6 Nb3 58 Bb5 Na5 At last, everything is ready! 59 Kg4! Nb7 60 Kf4 Nc5 61 a5 d3 62 Ke3 d2 63 Be2 Be2-d1 will follow, to prevent the fork on b3, then Ke3xd2 so Black resigns.