Monday, January 21, 2008

Endgame Caveat #3 Bahr's Rule

Bahr's Rule is a shortcut that allows you to evaluate some long pawn endings without actually going through long calculations. It involves the situation where blocked rook pawns are on one side of the board while the kings and one additional pawn are on the other side of the board.

It is somewhat difficult to write Bahr's Rule in a succinct manner, but here's my attempt. From the DEFENDER's blocked rook pawn, draw a line diagonally backward (toward the defending player's seat) to the nearest bishop file. Once at the bishop file, continue the diagonal line forward diagonally to the edge of the board. If the ATTACKER's extra pawn is ON OR BEHIND this line, then he wins. If the pawn is in front of the line, it's a draw.

Here's the typical situation where the blocked pawns are both on their respective fourth ranks.

The line of Bahr's Rule has been drawn and the h2 pawn falls on the line. This means that White can win this ending. The line goes: 1.Kg1 Kh4 2.Kg2 Kg4 3.Kf2 Kh3 4.Ke3 Kxh2 5.Kd4 Kg3 6.Kc5 Kf4 7.Kb5 Ke5 8.Kxa5 Kd6 9.Kb6 Kd7 10.Kb7 and the a-pawn will walk right in to a8.

Here's a less typical situation where the attacking pawn is blocked on its third rank.
The same method of drawing the line applies. Here the h-pawn is ahead of the line, so the result is a draw. e.g. 1.Kg1 Kh4 2.Kg2 Kg4 3.Kf2 Kh3 4.Ke3 Kxh2 5.Kd4 Kg3 6.Kc4 Kf4 7.Kb4 Ke5 8.Kxa4 Kd6 9.Kb5 Kc7 10.Ka6 Kb8 and White can't win.

How about the situation where White's pawn is on the fifth rank?

This is a trick question because you may draw the line, but it means nothing. White wins almost no matter where the pawn is because he has enough lead on the Black King to grab the important b7 square right after capturing on a6. This means that if the defender is to have any hope of defending this kind of ending, he needs to stop the rook pawn on the fourth rank. If the rook pawns aren't blocked yet then a4 should be met by a5.

There are a few more caveats mainly concerning situations when the extra pawn is very close to the blocked rook pawns or when one king is out of position, so remember that Bahr's Rule is a guideline that can accelerate your calculations, but you should double check through concrete analysis if you have time.

The following game occurred in Round 1 of the Class C Championships.

61...Ke7? Black chose to give up b4 in order to eliminate e6, an even trade, but why give up b4 if you don't have to? The defender of e6 is nowhere in sight. 'Do not hurry' - Shereshevsky. [ 61...a5 62.Kf1 Ke7 63.Ke2 Kxe6=] 62.Bxb4+ Kxe6 63.Kf1 Ke5 64.Ke2 Ke4 65.g3

[ Following the adage, "When behind, trade pawns not pieces," better would have been 65...f4!? 66.gxf4 Ne3 67.h5 Nf5 68.Kd2 Kxf4 69.Bc5 Kg5 70.Kc3 Ng3 71.Kb4 Ne4=] 66.Bd2 Nh5 Black flirts with being dominated. A knight on the rim is dim, but a dominated knight on the rim has three hooves in the grave. 67.Kf2 Kd3? [ 67...Nf6 going back to g4.] 68.Bb4?! [ 68.Bc1!
Kc2 ( 68...Kc3 preventing Bb2 but allowing 69.Kf3 Ng7 70.Bg5 Kc4 71.Bf6 Ne6 72.h5+-) 69.Bf4 Kb2 70.Be5+ The knight is dominated. Giving Black the passed a-pawn is not to be feared. 70...Kxa2 71.Kf3 Kb3 72.g4 fxg4+ 73.Kxg4 Kc4 74.Kxh5+- White's h-pawn will queen easily, while the White Bishop prevents Black's a-pawn from ever ascending to royalty.] 68...Ke4 69.Be7? [ 69.Bc3! Domination.] 69...Ng7 70.Bf6 Nh5? Allowing domination again. 71.Bh8

Between the White Bishop and the White King, the Black Knight can't get out to participate in the game any more. If the Black King moves to e6 to help the knight get out at f6, White simply exchanges into a won pawn ending. The endgame is already lost for Black with best play. 71...f4

White blew his last chance for a win with 72.gxf4?= [ 72.g4! Ng3 73.h5+-] 72...Nxf4 73.Kg3 Kf5 [ If Black is confident of his endgame knowledge, he can go for 73...Ng6 74.Bb2 Nxh4 75.Kxh4 Kd5 Once Black lodges his king at a8, he doesn't need the knight or the a-pawn to draw the the ending against Bishop and wrong rook Pawn. The Bishop cannot dig the king out of the a8 corner as it could for an h-pawn heading toward the h8 corner.] 74.Kf3 Ng6 75.Bc3 Nxh4+ 76.Ke3 Ng2+ 77.Kd4 Nf4 78.Kc5 Ke4 79.Bd2 Nd3+ 80.Kb6 Kd5 81.a4 Nb2 82.Kxa6 Nxa4 Drawn since neither side has mating material 1/2-1/2

In the post-mortem, the question came up, could Black hold the ending from this position with 71...Kd5? 72.Kf3 Ke6 73.g4 fxg4+ 74.Kxg4 Nf6+ 75.Bxf6 Kxf6 76.a4 a5

We have blocked rook pawns and an extra pawn. We could apply Bahr's Rule. The pawn seems to be ahead of the line implying a draw, but the king position here is important. Under normal circumstances in applying Bahr's rule, the defending king can only be pushed around so that it is two moves away from capturing an abandoned h-pawn. Here Bahr's Rule is subject to a caveat because Black's King is three moves away from capturing h4. If it were Black's move from this position, Kg6 would draw. 77.Kf4 Kg6 78.Ke5 Kh5 79.Kd5 Kxh4 80.Kc5 Kg5 81.Kb5 Kf6 82.Kxa5 Ke7 83.Kb6 Kd8 84.Kb7+- So the answer is No, Black was still lost after 71...Kd5.

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