Thursday, 20 October 2016

002. Skirting the Troitsky Line

Black: juliangon - thematic tournament,, 2016

Alexei Alexeyevich Troitsky (1866-1942) was a Russian analyst and study composer. One of his achievements was his definitive investigation of the endgame: king and two knights vs. king and pawn.

As we learn early on, two knights can't force mate against a bare king because stalemate always comes first. But give the defending side a pawn as well and stalemate doesn't apply. In that case it all depends how far the pawn has advanced, whether it can promote before the knights are ready to give checkmate. And how far is far enough depends upon which file the pawn is on. Troitsky worked out that if the pawn is blockaded by a knight no further forward than the fifth rank (RP, QP, KP), fourth rank (BP) and third rank (NP), then the knights have time to mate a cornered king. The imaginary pattern drawn across the board by this theoretical pawn is known as the Troitsky Line.

The tablebases now have five-piece (and six- and seven-piece) endgames worked out to perfection. Nevertheless, they have been unable to improve significantly on Troitsky's original analysis, which, according to John Nunn in Secrets of Minor-Piece Endings (Batsford 1995) was “astonishingly accurate”. Astonishing is the right word. Primarily because, given that one of the knights is stuck on blockading duty, how on earth do you drive the enemy king into a corner with just king and knight? Why doesn't it keep getting away?! Alexei Troitsky, presumably by means of extremely complicated retro-analysis, managed to prove that, while the king does keep getting away, in the final event it finds itself in a situation from which it can't escape.

Well, that was two knights vs. a single pawn. How about three pawns? I reached this ending in a recently finished online game. R+3P vs. 2N+3P eventually became 3P vs. 2N, at which point the Lomonosov endgame tablebases declared mate in 103. Fascinating!

It took me quite some time to understand why I was losing.

In the diagram Black has just played 57...Kf3! (the unique winning move). White can't jettison the a- and c-pawns because the h-pawn is much too far back. In fact doing so sees the white king corralled very quickly: 58 a6 Nxa6 59 c7 Nxc7 60 Kh4 Ne6 61 h3 Ng8 62 Kh5 Ne7 and mate in 10. So the game continued 58 Kh4 Kf4 59 Kh3 Na6! 60 Kg2 Kg4 61 h4 Ne8! (not 61...Kxh4? 62 c7 and draws) 62 h5 Kxh5, and now if 63 c7 then 63...Nexc7! and Black wins because the remaining white pawn is safely blockaded behind the Troitsky Line. The tablebase declares mate in 90.

Okay, we could have played that out, but there seemed little point. We were only continuing at all because the endgame was so unusual. So I found a different way to lose, one involving two little "traps": 73 Ka7!? (the "best" move 73 Kc8 leads to mate in 65), when 73...Kxc6? (or 71...Kxc6?) is stalemate, while 73...Kxa5? is a draw by Troitsky. The way Black solves this problem is quite nice: 73...Ne8! 74 Kb7 Nd6+ with mate in nine.

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