01 September 2011

Simple Positions, Pretty Geometries

In my latest post, A Slippery Opening, I commented on the opening from one of my recent games; in this post I'll give one of my recent chess960 endgames. I had Black in the diagram shown below.

When I first reached the position in my analysis I quickly concluded that, with two extra Pawns, Black wins easily. Black forces White to sacrifice the Bishop for one of the Pawns and wins with the other Pawn. After I played into the variation and received my opponent's reply, I looked at the position a second time and realized the position was not as straightforward as I had initially thought. If White can sacrifice the Bishop for the d-Pawn, the promotion square for the h-Pawn is not the color of Black's Bishop, so Black can't win.

Black to move

After 1.Bf6

After a more rigorous analysis I determined that the position is nevertheless won for Black. The d-Pawn must pass through two squares controlled by White's Bishop -- d2 and d4 -- where Black uses simple maneuvers of his own Bishop to shoo the enemy Bishop off the diagonal.

When we reached the diagram I played 1...Bb4, and the game continued 2.Kf5 Bc3 (first Bishop opposition) 3.Bg5 d4 4.Ke4 d3 5.Ke3 Be1 6.Kf3 Kc3 7.Kg4 Kc2 8.Bf4 Bd2 (second Bishop opposition). White resigned here because of 9.Bd6 Bc1 10.Bb4 Bb2, and with 11...Bc3, Black clears a diagonal for the last time. During the game I was more concerned with 2.Kh6, picking up the second Pawn. Here I calculated a maneuver resembling what happened in the game: 2...Bc3 3.Bg5 d4 4.Kxh7 d3 (I'll come back to this position in a moment) 5.Bf4 Kb3 6.Kg6 Kc2 7.Kf5 Bd2 8.Bd6 Bc1 9.Bb4 Bb2 10.Be1 Bc3.

After the game, I started to wonder how far my opponent was from achieving a draw. After 4.Kxh7 d3, the d-Pawn has only one Black square to cross and the King on h7 is too far from the action. On what squares would the White King have to stand in order to draw?

Since there are only five pieces on the board, a tablebase can answer that question. It turns out that if the White King is on g6 instead of h7, White to move draws, while Black to move wins the same way as in the game. Ditto for King on f7. The sequence to achieve the draw from g6 is 1.Kf5 Kb3 2.Ke4 Kc2 3.Kd5 (only move) 3...Bd2 4.Be7 Bh6 5.Bb4 (only move) 5...Bg7 6.Kc4. The White King arrives on c4 just in time to prevent the Black Bishop from moving to c3. This raised another question.

Since the White King started on g6, which is four moves from c4, why does the same result hold for the King starting on f7, which is only three moves from c4? Even with Black to move, White should have enough time to reach c4. The reason is that Black has a different winning sequence: 1...Kd5 (or 1...Kd4; other moves only draw) 2.Ke7 Ke4 3.Kd7 (not 3.Kd6; do you see why?) 3...Kf3 4.Kd6 Bd4 5.Bd2 Ke2 6.Bb4 Bf2 7.Ke5 Be1. Black had just enough time to oppose Bishops on the other side of the Pawn, where the White King can't help.

Sometimes the simplest positions contain the prettiest geometries.

No comments: