Example 4

Number of pieces: 11. Execution time: 4 hours.

FinalGen reaches its maximum efficiency when solving pawn endings, especially when the pawns are blocked or advanced.

In this ending, g4 and b7 are critical squares. If the white king can occupy one of them, White will be able to force the win.

If the white king moves to f3, then the black king must move to g5 in order to prevent White from occupying the key square g4 (moving to h5 allows the white king to reach b7.) So f3 and g5 are corresponding squares.

If the white king goes to a6, then the black king must go to b8. Moving to a8 would be a mistake, allowing the white king to get to g4 key square.

When the white king is on e2, the black king should be only on f6, and when the white king is on f2, the black king should be only on g6.

The win requires a neat little triangulation maneuver:
1. Kf1 Kg5
If 1... Kg6 then 2. Kf2 Kf6 3. Ke2 Ke7 4. Kf3 Kf6 5. Kg4.
2. Ke1 Kg6
2... Kf6 3. Ke2 Ke7 4. Kf3 Kf6 5. Kg4.
3. Kf2 Kf6 4. Ke2 Ke7 5. Kf3 Kf6 6. Kg4 1-0.

Example 5

Number of pieces: 8. Execution time: 1 hour and 2 minutes.

We also let Fritz 11 do an infinite analysis on this position. After letting the program run for over 1 hour, Fritz gave the following evaluations:
Kb3 +0.70
Ka3 +2.53
f6 +3.57
Kb4 +3.57
Kb5 +3.96
h5 +4.80
f5 +5.42

In this particular case, Fritz found the right move (Kb3) but its evaluation is not accurate enough to draw really valid conclusions.

A possible continuation will proceed:
1... Kb3 2. Kd2 a4 3. Kc1 Kc3 4. f4 Kd4 5. Nh5 Ke4 6. Kb2 Kf3 7. Ka3 Kg2 8. Kxa4 Kxh3 9. Kb4 Kg4 10. Nf6+ Kxf4 11. Kc4 Kf5 12. Ng8 h5 13. Kd3 Kf4 14. Ke2 Kg3 15. Kf1 ½-½.

Example 6

Number of pieces: 7. Execution time: 1 hour and 31 minutes.

The study below is by Kasparyan (first published in Bolshevistskaia smiena, 1928). White is to play and win.

1. d6 Be5 2. d7 Bc7 3. Kb5 bxa6+
White's task is easier after 3... Ka7 4. Bc5+.
4. Kc6
White does not fall into the trap. If 4. Kxa6? then 4... Kb8 5. Bd6 Ka8 6. Bxc7 stalemate.
4... Ba5 5. Bf6 Ka7 (5... Kb8 6. Be5+ Ka8 7. Bc7) 6. Bd4+ (if the pawn a6 had not existed, Black could now play 6... Ka3) Ka8 7. Bb6 1-0.

Fritz 11 finds the right move in this position, but it considers 4. Kxa6? (which only leads to a draw) to be a clear win (+2.92) for White.

Example 7

Number of pieces: 8. Execution time: 2 hours and 35 minutes.

Kasparyan is also the author of the following composition:

1. Kd4
Black was threatening 1... b3. 1. Bc2 fails because of 1... Kc7 2. Kd4 Kd6 3. Bd3 a3 4. Bc2 Bc8 and 5... Be6.
1. Kd6 is not enough. Black wins after 1... b3 2. a3 b4 3. axb4 b2 4. Bc2 a3.
Remark: Regarding the variation above, the author considers 2... b2 (instead of b4) to be winning for Black, but after 2... b2 3. Bc2 b4 4. Kc5 axb3 5. Bb1 the game is a draw.
1... b3
Other attempts to improve the location of the black pieces fail:
1... Bc8 2. Bc2 Be6 3. Kc5 ½-½.
1... Kc7 2. Kc5 b3 3. a3 Bc8 4. Kb4 Be6 5. Kc3 Kd6 6. Bh5 Bc4 7. Be8 Kd5 8. Kb2 Kd4 9. Bd7 ½-½.
2. Kc3
All other moves lose:
2. axb3 a3 3. Kc3 b4+ 4. Kc2 Bd3+ 5. Kc1 Kc7 followed by Kb6-c5-d4-c3.
2. a3 b4 3. axb4 b2 4. Bc2 a3 5. Bb1 (5. Kc3 Bc4 6. Bb1 Be6) 5... Bc8 6. Kc3 Be6 7. Bd3 Kc7 8. Bb1 Kd6 9. Bc2 Ke5 10. Bb1 Bd5 11. b5 Kd6.
2... bxa2
If 2... b4+ 3. Kb2 bxa2 (3... Bc4 4. a3 ½-½) 4. Bxa4 ½-½.
3. Kb2
White is now threatening 4. Bxa4.
3... a3+
After 3... a1=Q+, White replies 4. Kxa1 a3 5. Be2 followed by Bxb5 and draws.
4. Ka1
4. Kxa2 would be a mistake because of b4 5. Bb3 Kc7 6. Bg8 Kb6 7. Kb3 Kc5.
4... b4
Now, three extra pawns are not enough to win. A possible continuation will proceed:
5. Bb3 Kc7 6. Ba4 Kd6 7. Bb3 Kc5 8. Ba4 Kd4 9. Bb3 Kc3 10. Ba4 Bf1 11. Bb3 Bd3 12. Ba4 Bc2 13. Bb3 ½-½.

It is interesting to notice that Fritz still evaluates the position as an easy win for Black. After 13. Bb3, it evaluates the position as -5.23 in Black's favour.

Example 8

Number of pieces: 8. Execution time: 6 hours.

This is a study by Selesniev. Since the pawns have very limited mobility in this position, FinalGen took 6 hours to solve it. Solving other eight-piece positions may take longer.

1. Kf7 Rxh5 2. Rg8+ Kh7 3. Rg7+ Kh8 4. Kg6 g4 5. Ra7 Rg5+ 6. Kxh6 g3 7. Kxg5 g2 8. Ra1 f3 9. Kg6 1-0.

Example 9

Number of pieces: 8. Execution time: 6 minutes.

In the following position White can get a draw by means of mate threats despite the fact that the black pawns are far advanced.

1. Kb6
If 1. Rd4, then 1... e2 (1... d2 2. Kb6 transposes into the main line) 2. Kb6 Kc8 0-1.
If 1. Re4 d2 (1... e2 2. Kb6 Kc8 3. Kc6 ½-½) 2. Kb6 Kc8 0-1.
1... d2 2. Rd4 Kc8 3. Kc6 e2 4. Ra4
The mate threat allows White to gain an important tempo.
4... Kb8 5. Rb4+ Ka7 6. Ra4+ Kb8 7. Rb4+ Kc8 8. Ra4 Kd8 9. Rd4+ Ke7 10. Re4+ Kf8 11. Rf4+ Kg8 12. Ra4 ½-½.

Example 10

Number of pieces: 7. Execution time: 5 minutes.

Before introducing the following example, let's take a look at a theoretical position by J. Walker.

In this ending, White wins regardless of the Kings location.

1. Ke4 Ke6 2. Kd4 Kd6 3. Kc4 Kc6 4. Bc5 Kc7 5. Kd5 Kb8 6. Kd6 Kc8 7. Bb6 Kb8 8. Kd7 Ka8 9. Bc7 b5 10. axb6 Kb7 11. Kd6 wins.

The example below (by Kasparyan) is inspired by the previous endgame.

1. Bf6
1. Ke3 e5 2. Bf6 b6 3. axb6 Kc6 4. Bd8 a5 draw.

Now there are two variations:

a)
1... Kc5
Now threatening b5 with a draw. Playing 1... b5 immediately would be weak because of 2. axb6 Kc6 3. Bd4 wins.
2. Bd8
White threatens Bb6 reaching the Walker position.
2... b6 3. axb6 Kc6 4. Ke3 a5 (4... e5 5. Ke4 a5 6. Kxe5 1-0) 5. Kd4 a4 6. Kc4 e5 7. Kb4 e4 8. Kxa4 e3 9. Ka5 e2 10. Bh4 wins.

b)
1... Kc6
Black does not waste any time, but now the bishop can comfortably occupy the d4 square.
2. Bd4 Kb5 3. Bb6
Black has reached the Walker position, with the addition of the pawn e6.
3... Kc6 4. Ke3 Kd6 5. Kd4
The white king marches (by the most direct route) to the pawn e6.
5... Kc6
White's task is easier after 5... Kd7 6. Kc5.
6. Ke5 Kd7 7. Kf6 Kd6 8. Kf7
White tries to provoke the e-pawn's advance.
8... Kd7 9. Kf8 Kd6 10. Ke8 e5
Finally, the pawn is forced to advance. If 10... Kc6, then 11. Ke7 Kd5 12. Kd7 e5 13. Kc7 Kc4 14. Kd6 e4 15. Ke5 Kd3 16. Kf4 1-0.
11. Kf7 e4 12. Kf6 e3 13. Bxe3 b6 14. axb6 a5 15. Bc5+ Kc6 16. Ke5 a4 17. Kd4 a3 18. Kc3 a2 19. Kb2 wins.

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