Monday, August 30, 2010

The Ending of Rook, Rook's Pawn, and Bishop's Pawn vs. Rook

Over the weekend I was going over some of my Internet Dragon games and I came to a five-minute game played on 3-2-08 vs. "duc". After 39 Rf7, we got to the following position: White: K on b1, R on f7, P's on a2 and c2; Black: K on g4, R on a8. Being two Pawns up, one would think White would win handily, but the game continued 1...Rb8ch 2 Kc1 Ra8 3 Kb2 Rb8ch 4 Kc3 Ra8 5 Kb3 Rb8ch 6 Ka4 Rc8 7 Rf2 Ra8ch 8 Kb3 Rb8ch 9 Kc3 Rc8ch 10 Kd2 Ra8 11 Rg2ch Kf3 12 Rg1 Ke4 13 Ra1 Kd4 14 a4 Kc4 15 a5 Kb5 16 c3 Rxa5 17 Rxa5ch Kxa5 18 Kd3 Kb5 and drawn in 25 more moves as he played it out to the bitter end.

When I first annotated this I made the comment that "White should have an easy win". After playing around with the position awhile, I crossed out the "n" and the "easy", and said simply that "White should win".

Then I decided to check my ending books, and found that all three said this ending was a book draw, although one commentator stressed that in practice White often wins because of imprecise play by Black. It bothered me, though, that all of the positions given had Black's King near the Pawns, and in none was the Black King cut off as in my game.

After stewing about this all Saturday night (literally, as I tend to wake up during the night and study chess positions in my head), I woke up yesterday morning and got the idea of ignoring the a-Pawn and treating it as Rook plus c-Pawn vs. Rook. The book learning on this is that it is a draw when the Black King can reach the Queening square, and a loss if it cannot. When the Black King is in front of the Pawn, it is called Philidor's Position, and when the Black King is cut off by a file from the Pawn, it is called the Lucena Position.

In the latter position, the White Rook does duty cutting off the Black King, while the White King attempts to escort the Pawn to the promised land. When the Pawn is on the 5th rank, this is easy. But when the Pawn is not yet on the 5th rank, it becomes quite tricky. If you play with the position you will see that the Black Rook stays in front of the Pawn, on its back rank if possible, i.e., he sets up camp on White's Queening square. Just as in my game, he then checks the White King whenever the King attempts to slide up the board beside the Pawn, hoping to support its advance.

It is therefore apparent that to achieve the Pawn's advance from the 2nd, 3rd, or 4th rank, the assistance of the Rook will be necessary. The Rook can assist either from the rank or from the file, behind the Pawn. When the Rook does this the Black King will then have time to come closer to the action, hence it must initially be cut off by *more* than the single file required when the Pawn is on the 5th rank. Fine, in "Basic Chess Endings", gives the rule as follows: "If the P is on the 3rd or 4th rank and its K is near it, White can always force a win if and only if the Black King is cut off at a distance of 3 files from the Pawn (NP), or 2 files from the Pawn (BP or center P). If the P is on the 2nd rank, and Black's K is on the 4th or 5th, White wins if and only if the K is cut off at a distance of 5 files from the Pawn."

It is obvious that with the c-Pawn it is impossible for the Black King to be cut off by 5 files. It is possible with the b-Pawn, and Fine gives this example, position 317: White: K on b1, R on f1, P on b2; Black K on h4 and R on b8. This illustrates the principle well, because (according to Fine) White to play wins, while Black to play draws; i.e., White to play can play 1 Rg1, cutting off the Black King by the required 5 files, while if it's Black's turn he plays 1...Kg4, getting one file closer to the action. The winning method with White to play runs 1 Rg1 Kh5 2 Kc2 Rc8ch 3 Kd3 Rb8 4 Kc3 Rc8ch 5 Kd4 Rb8 6 Rb1 (the Rook slides over behind the Pawn) Kg6 7 b4 (White gets the Pawn pushed finally) Kf7 (Black gets another file closer) 8 Kc5! Ke7 (8...Rb8 does not help, as 9 Kd6 Rb8 10 Kc7 Rb5 11 Kc6 allows the Pawn to advance next move, where it will be on the 5th with an easy win) 9 Kc6! winning. If 9 b5? Kd7 and the Black King will get to the Queening square and draw. But after 9 Kc6!, White keeps the King out with a little trick: 9...Kd8 10 b5 Kf8? 11 Rh1! and Black either gets mated or loses his Rook.

Euwe and Hooper, in "A Guide to Chess Endings", give the exact same position as the Fine position just discussed, but they give it a little different treatment. Their winning line with White to move is 1 Kc2 Rc8ch 2 Kd3 Rb8 3 Kc3 Rc8ch 4 Kd4 Rd8ch 5 Kc5 Rc8ch 6 Kd6 Rb8 7 Rb1 (The White Rook slides over behind his Pawn) Rb3 8 Kc5 Kg5 9 Kc4 Rb8 10 b4 winning.

Euwe and Hooper point out that in the initial position White's simplest win is to cut off Black's King on the rank with 1 Rf5 Kg4 2 Rc5 Kf4 3 Kc2 Ke4 4 Kc3 winning as b4 follows.

Euwe and Hooper's discovered that Fine was wrong in saying Black draws if he moves first in the initial position. They give 1...Kg5 (Not 1...Kg3 2 Rf5, and not 1...Kg4 2 Rf6 Rh8 3 Ra6 ) 2 Rf2 Kg4 3 Kc1!! This discovery they atribute to Kopaiev, the idea being it puts Black into zugzwang. Both Black's King and Rook are on their best squares, and Black loses because he must move one of them!! Now if 3...Rc8ch 4 Rc2 Rh8 5 Rc5 Rh2 and White wins by moving his K to a3 and then advancing the Pawn to b4. After 3 Kc1!! Euwe and Hooper's main line goes 3...Rh8 4 b3 Kg3 5 Rf6 Rh2 6 b4 Kg4 7 b5 Kg5 8 Rf8 Rh7 9 Rb8 Kf6 10 b6 Ke6 11 b7 and they say wins though I don't understand why Black's King can't simply move in for the draw.

But now Euwe and Hooper give an example with the c-Pawn. Position is White: K on c1 R on g1, and Pawn on c2; Black: K on h5 and R on c8. Very similar to the last position but on the c-file rather than the b-file. They say White wins, contradicting Fine on the number of files the Black King needs to be cut off (only 4 here). Here the White King moves up on the short side, which wasn't there for the b-Pawn as there was only one file on the "short side" there. The main line runs 1 Kb2 Rb8ch 2 Ka3 Rc8 3 Kb3 Rb8ch 4 Ka4 Rc8 5 Rc1 (the usual "sliding over" of the White rook to support his Pawn's advance) Kg6 6 c4 Kf6 7 Kb5 Rb8ch 8 Kc6 Ke7 9 Re1ch Kd8 10 Rd1ch Ke7 (10...Kc8? 11 Rh1 threatening Rh8) 11 c5.

In my game,. however, the King is cut off by "only" 3 files, so White cannot win using this method. However, there has to be a way to win using fact that he has the extra pawn on the a-file. This requires further study.

No comments: