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Two evenly matched players have reached the endgame exchanging off all their pieces except the King and 3 Pawns. Having castled on opposite sides, we have now reached the following interesting position.

8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1

Quite clearly, this is now a race for pawn promotion. What should be the strategy in this position (for both players)? Should your King go after the enemy pawns, or lead your own pawns to promotion?

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Good question. Draw I suppose. –  Zistoloen Dec 7 '12 at 13:08
    
Who has the first move here? –  xaisoft Dec 7 '12 at 17:19
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The position is symmetrical so it doesn't matter. Let's just say white has the first move. –  Happy Dec 7 '12 at 17:23
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Well if it is white's first move and both players just move their outside pawns for promotion, white will win. If it is black's first move, black will win. This would mostly likely be the course of play though. –  xaisoft Dec 7 '12 at 17:27
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For these types of positions, I play the computer against itself. When it's done, I look at the moves to find those with which I disagree. Then I go back, and force the moves I like to see how the computer punishes me. This helps me understand why the computer made the move it made. It's like having an instructor. –  Tony Ennis Dec 8 '12 at 14:38

3 Answers 3

up vote 10 down vote accepted

The two previous answers both suggested that this position is a draw, but I believe it's a win for White.

Stockfish 5 evaluates the position at more than +10 if left to run for long enough (e.g., to depth 40), which sounds pretty decisive, but of course Stockfish isn't infallible. I generated a FinalGen tablebase for the position, but unfortunately it wasn't able to solve the position fully, though it did show that the moves 1.Kf3 and 1.Kf1 lose. Stockfish evaluations for the positions after 1.Kf3 and 1.Kf1 eventually drop below -10, which suggests that Stockfish does understand what is going on here, and its evaluations (at depth 40 or more) should be taken seriously.

The question, of course, wasn't about the game-theoretic status of the position, but about the strategy. The basic strategy is to block the opponent's pawns and aim for zugzwang. White's first move appears to give him a decisive advantage in doing this, provided he starts by moving his king towards the queenside (1.Kd1, 1.Kd2 or 1.Kd3). Here's a sample line:

[FEN "8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1"]

1.Kd3 Ke6 2.h4 Kf7 3.h5 Kg7
( 3...a5 4.Kc4 a4 5.Kb4 b5 6.f4 c5+ 7.Ka3 c4 8.Kb4 Kg7
( 8...Kf6 9.g4 Kg7 10.f5 )
9.f5 Kg8 10.g3 Kg7 11.g4 Kg8 12.h6 Kh7 13.g5 Kg8 14.g6 Kf8 15.f6 $18 )
4.g4 Kh6 5.Kc3 a5 6.f4 a4 7.f5 Kg5
{ The kingside is now mutual zugzwang: Black can't move there without allowing promotion, and White can't move there without losing a pawn. Play therefore switches to the queenside. }
( 7...b5 8.f6 $18 )
8.Kb4 b5 9.Ka3 c5 ( 9...c6 10.Kb4 c5+ 11.Ka3 c4 12.Kb4 $23 { Zugzwang. } )
10.Ka2 b4 11.Kb2 a3+ 12.Kb3 c4+ 13.Ka2 c3 14.Kb3
{ The queenside is now also mutual zugzwang, so Black to move loses. }
14...c2 15.Kxc2 a2 16.Kb2 b3 17.Ka1 Kf6 18.h6 Kf7 19.g5 Kg8 20.g6 Kf8 21.f6 1-0

Black could try to avoid zugzwang by mimicking White's moves, but he can't mimic checks, and this is in any case a recipe for losing a direct race to promote. Here's a sample line:

[FEN "8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1"]

1.Kd3 Ke6 2.h4 a5 3.Kc4 Kf5 4.h5 a4 5.g4+ Kg5 6.Kb4 b5 7.f4+ Kh6 8.f5 c5+
9.Ka3 c4 10.f6 c3 11.g5+ Kxg5 12.f7 b4+ 13.Ka2 b3+ 14.Kb1 a3 15.f8=Q a2+
16.Ka1 c2 17.Qc5+ 1-0
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So you're saying the initial position is already a black-losing zugzwang, given proper play ? In the first diagram, 7... Kg5 looks like a blunder, what if B just pushes pawns ? Is 9... c5 and c6 the same ? –  Nikana Reklawyks Nov 17 at 6:09
    
@Nikana: In the initial position, the player to move wins, so it's not zugzwang. In the first diagram, 7...Kg5 is necessary in order to stop the white pawns continuing to advance - otherwise essentially the same thing that happens after 17...Kf6 would happen immediately. 9...c6 doesn't seem to help - see the variation I've added to the diagram. –  Stephen Nov 17 at 9:12
    
Looks pretty sound, +1. What if (first diag) B (or any player really) rushes pawns ahead, instead of "losing time" with Kg7. The goal would be to force the other's king not to move, while having one's own king free to lose tempi (e.g. Kh7). Of course white should win at this game, but maybe both kings only must not commit themselves into the f5 g4 h5 formation. –  Nikana Reklawyks Nov 17 at 13:13
    
@Nikana: I've added a variation in which Black refrains from 3...Kg7 and instead starts pushing pawns. This line results in mutual zugzwang on the queenside, but Black is then unable to stop White's pawns. –  Stephen Nov 17 at 14:55

I used to think the following draw by repetition looked optimal for complicated reasons (detailed below), but now I doubt it. White may well have a winning strategy, involving putting Black in zugzwang on both sides. See Stephen's answer.

[FEN "8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1 "]

1. h4  a5  2. h5 Ke7 3. h6 Kf6 4. g4 a4 5. Kd2 Kg6 6. g5 a3 7. Kc2 b5 8. Kb3 b4 9. Ka2 Kh7 10. Kb3 Kg6 ½-½

Neither player can advance the ♟c/♙f, or will end up in zugzwang.

If White advances the ♙f, Black wins.

[FEN "8/2p5/6kP/6P1/1p6/pK6/5P2/8 w - - 8 9"]

  9. f3 c6 10. f4 c5 (9. f4 c5) 11. f5+ Kh7 12. Ka2 (12. f6 Kg6 -+) c4 13. Ka1 b3 14. Kb1 c3 15. Ka1 c2 (13. Kb1 b3 14. Ka1 c3 15. Kb1 b2 16. Ka2 c2 17. Kxa3 c1=Q -+(16. Kc2 a2 17. Kxc3 a1=Q -+)) 16. g6+ Kxh6 17. g7 c1=Q# (15. Kc1 a2 -+)

If Black advances the ♟c, White wins.

Notice how it falls back to White playing ♙f4 with reverse colors.

[FEN "8/2p5/6kP/6P1/1p6/pK6/5P2/8 w - - 8 9"]

  9. Ka2 c6 10. f4! c5 (9... c5 10. f3! c4 11. f4 +- see Wf4) 11. Kb3 c4+ 12. Ka2 +-

I wouldn't call that a full analysis, especially as the first moves are played so hastily, but I don't think neither player is in zugzwang from the start, as back-and-forth moves on columns seem free.

As neither the rush of pawns work, nor helping one's pawns with one's king (which must stay in the opponent's far-pawn's square), I think the position is symetrical enough to be a fair draw.

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1  
Hooray for the replayer in which this was carried on. –  Nikana Reklawyks Dec 8 '12 at 11:59
    
I think the key is both players must keep their "inside pawns" back; it took me a while to see that. It is very tempting to try to build a Phalanx(f4,g4,h4), which I believe is actually losing. –  Akavall Dec 8 '12 at 16:15
    
I think all of this analysis is correct, except for the very first move: 1.h4 throws away the win. –  Stephen Nov 21 at 19:17
    
@Stephen: You may very well be right, emphasizing "as the first moves are played so hastily". However, you show a (supposedly winning) line going 1. Kd3 Ke6 2. h4 : could you explain how that is different from 1. h4 … Ke7 … Kd2 ? –  Nikana Reklawyks Nov 21 at 20:44
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@Nikana: The important difference is the move order (the difference in the king moves is not very relevant: 1.Kd2 Ke7 2.h4 seems similar to 1.Kd3 Ke6 2.h4). The problem with starting with 1.h4 is that Black can play 1...a5, and White is no longer a tempo ahead on the queenside, which seems to make it impossible for White to put Black in zugzwang. In particular, after 1.Kd3 a5 (or 1.Kd2 a5) I think White needs to immediately move his king to the c-file, whereas after 1.h4 a5 he can't. –  Stephen Nov 21 at 22:13

Unless there is careless pay, I see this ending up in a draw, play might continue:

[FEN "8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1"]

1. Kd3 Ke6 2. g4 a5 3. h4 a4 4. h5 Kf6 5. Kc3 b5 6. h6 Kg6 7. g5 a3 8. Kb3 b4 9. Ka2 Kh7
10. Kb3 Kg6 11. Ka2

Black can't do 11... c5, because then it is basically over with:

  [FEN "8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1"]

 1. Kd3 Ke6 2. g4  a5  3. h4  a4  4. h5  Kf6  5. Kc3  b5 
 6. h6  Kg6  7. g5  a3 8. Kb3  b4  9. Ka2  Kh7  10. Kb3  Kg6 
 11. Ka2  c5  12. f3 c4  13. f4  c3  14. Kb3  a2  15. Kxa2  c2 
 16. Kb2 b3 17. Kc1  Kh7 18. f5  Kg8  19. f6  b2+  20. Kxc2  Kf7 
 21. h7  Kg6 22. h8=Q
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Errr, most of the moves you show in the second diagram are totally unenderstandeable if you don't explain the underlying zugzwang a bit more. What about 11... c6, if each and every tempo are important ? If White f4, then he's the screwed one. (Unfortunately, I can't carry the analysis myself.) –  Nikana Reklawyks Dec 7 '12 at 18:43
    
If black plays 11...c6, white should still play 12. f4. If white, however, plays 12. f3, then he is screwed. Going back to 11...c6, it could continue 12. f4 c5 13. f5+ Kxf5 14. h7 Kxg5 15. h8=Q" –  xaisoft Dec 7 '12 at 18:53
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13... Kh7 makes more sense. –  Nikana Reklawyks Dec 7 '12 at 19:40
    
Whoever downvoted? An explanation would be nice? –  xaisoft Dec 8 '12 at 13:06
    
Why is this downvoted? I don't see a difference between @NikanaReklawyks post and this except some more detail in theirs. –  xaisoft Dec 9 '12 at 18:31

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