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.