user587: Two evenly matched players have reached the endgame after 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? 

user672: 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.

user4182: 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


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

<pre><code>8/pppk4/8/8/8/8/4KPPP/8 w - - 0 1
</code></pre>

<p>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? </p>
 672: <p>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 <a href="https://chess.stackexchange.com/a/7822/672">Stephen's answer</a>.</p>

<pre><code>[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
</code></pre>

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

<h1>If White advances the <code>♙f</code>, Black wins.</h1>

<pre><code>[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 -+)
</code></pre>

<p></p>

<h1>If Black advances the <code>♟c</code>, White wins.</h1>

<p>Notice how it falls back to White playing <code>♙f4</code> with reverse colors.</p>

<pre><code>[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 +-
</code></pre>

<p>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 <em>from the start</em>, as back-and-forth moves on columns seem free. </p>

<p>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.</p>
 4182: <p>The two previous answers both suggested that this position is a draw, but I believe it's a win for White.</p>

<p>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.</p>

<p>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:</p>

<pre><code>[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
</code></pre>

<p>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:</p>

<pre><code>[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
</code></pre>
