26 September 2007

K + 2N vs. K + P

The strange looking diagram shown below is a continuation of Tablebase 1 - Botvinnik 0. There I pointed out that Botvinnik had incorrectly analyzed a position in an endgame of two Knights against a Pawn. The simpler endgame of two Knights without other pieces is a theoretical draw, because the Knights can't contruct a mating position without first stalemating the enemy King. The extra Pawn allows the Knights to play through the 'stalemate' position because the weak side has a Pawn move and isn't stalemated. It is the trickiest of the 'elementary endgames'.

Not all positions with 2N vs. P are won for the strong side. Ignoring for a moment the blue symbols, an example is the diagram. The position of the five pieces, with White to move, is a draw. It is an exception to a clever, but faulty, schema developed by the famous endgame composer Alexey Troitsky (1866-1942). Botvinnik relied on the schema to reach his erroneous conclusion that the 1941 game Smyslov - Lilienthal was a win for Black.


White to Move

Looking at the diagram, I started to wonder how sensitive the result was to the position of the Knight on c2, the only piece that has real freedom of movement. The Knight on h7 is tied down to blockading the Pawn until the critical moment and the Black King must stay near the White King to restrict its space.

Keeping the position of the other pieces fixed, always with White to move, I shifted the Nc2 to its available squares and recorded the results. A square marked 'D' means the position where the Knight starts on that square is a draw. For example, placing the Knight on h4 is an obvious draw, because White plays Kxh4. A square marked 'S' indicates an immediate stalemate, with no further play possible.

The squares marked with a number show the number of moves required to win when the Knight starts on that square. For example, shifting the Knight from c2 to g5 allows Black to win in eight moves: 1.Kh4 Kf4 2.Kh5 Kg3 3.Kg6 Kg4 4.Kg7 Kf5 5.Kh8 Kf6 6.Kg8 Kg6 7.Kh8 Nf6 8.h7 Nf7 mate.

The longest win, with the Knight starting on e3, takes 54 moves. It goes like this: 1.Kh4 Kf4 2.Kh5 Nd5 3.Kg6 Ndf6 4.Kf7 Kf5 5.Ke7 Ke5 6.Kd8 Kd6 7.Kc8 Kc6 8.Kd8 Nd7 9.Kc8 Ndf8 10.Kb8 Ne6 11.Kc8 Kc5 12.Kb7 Kb5 13.Ka8 Ka6 14.Kb8 Kb6 15.Kc8 Kc6 16.Kb8 Nc5 17.Ka7 Kc7 18.Ka8 Kb6 19.Kb8 Nd7+ 20.Kc8 Kc6 21.Kd8 Nb6 22.Ke8 Kd5 23.Ke7 Ke5 24.Ke8 Ke6 25.Kd8 Kd6 26.Ke8 Nc8 27.Kd8 Na7 28.Ke8 Nc6 29.Kf7 Kd7 30.Kg7 Ke7 31.Kg6 Ke6 32.Kh5 Kf5 33.Kh4 Kf4 34.Kh3 Kf3 35.Kh2 Kf2 36.Kh3 Ne5 37.Kh4 Kg2 38.Kh5 Kf3 39.Kh4 Nf7 40.Kh3 Nfg5+ 41.Kh2 Kf2 42.Kh1 Ne6 43.Kh2 Nf4 44.Kh1 Kg3 45.Kg1 Ng2 46.Kf1 Kf3 47.Kg1 Ne3 48.Kh2 Kg4 49.Kg1 Kg3 50.Kh1 Kf2 51.Kh2 Ng5 52.Kh1 Ng4 53.h7 Ne4 54.h8=Q Ng3 mate. This variation shows best play for both sides, although there are branches of equal value at many points.

There are several mechanisms at work in the different solutions. Together they show how the simplest chess positions can illustrate attractive geometric patterns. • Reference: Shredder 6 piece tablebase.

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Note: For the record, I submitted this to Chess Blog Carnival II, but it wasn't used.

Later: After Jack Le Moine explained that my submission had been lost, not rejected, I tried again. Second time lucky: November Carnival of Chess Blogs.

1 comment:

Ryan Emmett said...

Absolutely fascinating. It's truly amazing how a seemingly simple position can create so much complexity. Thanks for posting.