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29 Feb

The Leaping Knight

Forgive my indulgence, but with today being 29th February, I only get this chance every four years to pay homage to those great leaper’s in chess, namely the knights - the only piece on the board that is allowed to jump over pieces and leap around the 64-squared board with a series of ‘L-shaped’ moves. And this in turn has made for an interesting, centuries-old mathematical conundrum.

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Arguably one of the oldest problems in chess is the “Knight’s Tour,” where you throw away all the pieces save for a lone knight.  Now place the knight on any one of the 64 squares of the chess board and see if you can make 63 legal moves so that you visit every square once, and only once. This problem has fascinated mathematicians and puzzle aficionados dating back to the Middle Ages - and some speculate it could even antedate chess, with the L-shaped knight move possibly deriving from the tour!

euler-8x8-knight-tour-magic-square.gif

The subject then seems to have been forgotten until around 1722. The first thorough scientific study of knight’s tours was conducted by Leonhard Euler in 1759; and many of his examples now appear in today’s classrooms as magic square challenges - and it has also been the subject of complex mathematical studies.  

Can you complete the knight's tour magic square opposite?

In 1995, mathematicians Martin Löbbing and Ingo Wegener wrote a paper with the appealing title: "The Number of Knight's Tours equals 33,439,123,484,294 -- Counting with Binary Decision Diagrams.” They obtained their result by running 20 Sun workstations for four months. Not to be outdone, two years later, Professor Brendan McKay of the University of Melbourne added to the debate by using another method (splitting the board into two halves) and got the result down to 13,267,364,410,532.

To give you an idea of the magnitude of this formulae for the large number of tour possibilities, a computer searching and finding them at a speed of one million per minute would need more than 25 years to calculate Professor McKay’s solution.

Czech grandmaster Vlastimilk Hort, a former candidate for the world title, once said that in blitz, the leaping knight was stronger than the bishop. And today’s game is one of my favourite blitz games that demonstrates just how effective a knight can be in blitz, as Garry Kasparov’s knight-pair traverse around the board before they deliver the equine killer blow.

GM Vladimir Kramnik - GM Garry Kasparov
Munich, 1994
King;s Indian Defence
1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.Nf3 0-0 6.Be2 e5 7.d5 a5 8.Bg5 h6 9.Bh4 Na6 This is a system against White's set-up that was refined by Soviet Leonid Stein, who died of a heart attack at the age of just 38 in 1973. 10.0-0 Bd7 11.Nd2 Nc5 12.b3 Nfxe4 The King's Indian Defence is all about tactics, tactics, tactics - and in a blitz game, Kasparov is alert to all the possibilities of confusing his opponent with such a bold tactic, unsound or otherwise. 13.Bxd8 Nxc3 14.Qe1 The only move, as 14.Qc2 Nxe2+ 15.Kh1 Nd4 and Black will have three excellent pieces for the queen. 14...Rfxd8 15.Rc1 Stronger was 15.f3 but Kramnik was probably concerned about 15...e4 and Kasparov opening up the position for all of his very active pieces. 15...Nxa2 16.Ra1 Nb4 The body count is that Black has two pieces and two pawns for the queen. Normally this would favour the queen, but here White's position is difficult and cramped, while Black is solid and he has lots of active piece-play potential - and note now how White's weak b3 pawn makes life difficult for Kramnik to unravel from this mess. 17.Bd1 e4 18.Rb1 Re8 19.Qe3 f5 20.h4 Rf8! Kasparov has the time and the luxury to place his rooks on their most optimum squares now of f8 and e8. 21.g3 Rae8 22.Kg2 Nbd3 23.Rg1 f4 24.gxf4 Rxf4 25.h5 g5 26.Rf1 Rh4 27.Rh1 Rf4 28.Rf1 Ref8! The attack on f2 is overwhelming, forcing Kramnik into his only option now. 29.f3 Rh4 Also winning was 29...Bd4 30.Qxd4 exf3+ 31.Nxf3 Rxd4 32.Nxd4 Bh3+ 33.Kxh3 Rxf1 and Black will be two pawns up in the ending. 30.fxe4 Nf4+ The clinical wins was 30...Rxf1 31.Nxf1 Rh3 32.Bf3 (32.Qxh3? Nf4+!) 32...Be5 33.Kg1 g4 - but Kasparov opts to prolong Kramnik's ‘knightmare'. 31.Kg1 Ncd3 32.e5 Kramnik can't defend against Kasparov's leaping knights and active pieces swarming around the White king. And the alternative didn't fair any better: 32.Nf3 Rg4+ 33.Kh1 Rg3 34.Qd2 Bh3 35.Rg1 Bg2+ 36.Rxg2 Rxg2 37.Qxg2 Nxg2 38.Kxg2 g4 39.Nh2 Ne1+! 40.Kg3 (40.Kh1 g3) 40...Be5+ winning a piece. 32...Nxe5 (See Diagram) 33.Rc1 Rh3 34.Nf3 After 34.Qa7 the knights quickly come in for the kill after 34...Rd3 35.Rc2 Bg4! 36.Bxg4 Nxg4 37.Nf3 Ne3 and White will suffer heavy material lose. 34...g4 35.Nxe5 Rxe3 36.Nxd7 Kramnik may have rid himself of one troublesome knight - but the other soon leaps in to deliver the fatal kick. 36...Nh3+ 37.Kg2 Rxf1 38.Kxf1 g3 39.Kg2 There's no hope. If 39.Bg4 Nf4 40.Rd1 Bd4! 41.Rd2 Rd3! 42.Rxd3 (42.Rg2 Kf7 43.Be6+ Ke7 44.Bg4 Rxb3 and either the a- or g-pawn will pass.) 42...g2+ 43.Ke1 g1Q+. 39...Nf4+ 0-1 If 40.Kf1 g2+ 41.Kf2 Bd4 42.Nf6+ Kf7 43.Bf3 Re1+ easily wins. 

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