Can White win?

You're right, and good catch - mimicking doesn't entirely work for white, since white isn't playing to draw. White needs to wait for the right opportunity to break the mimickry.

After 3...Kf7, white can play 4. Ke2 and 5. Kd3 no matter what black does, since as other people pointed out, black needs to answer Kd4 with ...Kc6 and Ke4 with ...Kd6.

After 5. Kd3:

• If the black answers with  5...Kd7, play the zugzwang move 6. Ke3! and you're back to mimickry.
• If black doesn't answer with 5...Kd7, black does't have enough time to answer 6. Kd4 with 6...Kc6. So 6. Kd4 wins.

EDIT: And yes, 3. Kd2 is also a good time to break the mimickry since you can answer 3...Kd6 with 4. Ke2! or 3...Kd7 with 4. Ke3!, preserving the zugzwang.

Been pushing pieces around on my board for a good hour & I've come around to the idea that white can win this but not in the middle of the board & only from one starting position.

If black makes the mistake of landing on c6 when white's king is on e4, then white has to set off up the h-file & it will arrive at h6 before black can stop it. Then it captures the f6 pawn & is able to promote the f5 pawn a full 2 moves ahead of black (or at least one).

If black goes after white's f4 pawn instead, then white can use the a5 pawn move at the last moment to switch tempo from black & still capture the f6 pawn.

If the two kings remain in the middle, then it's a draw.

Here's my poor attempt at a diagram:-

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It all depends on which square the black king is sitting: c6 & black loses the game but d6 & it can force a draw. The critical square for white to reach is h6.

From your diagram, 1.Kf3 Kd5! 2.Kg4 Ke4 =

I think I see what you're getting at ChaoChinKun, I was assuming black would take f4 but of course it will take f5 if white is careless - thanks for pointing that out.

Here's how I think it should be done:-

White king on e4, black on c6

1.Kf3  Kd5
2.Kg3  Ke4
3.Kg4  Ke3
4.Kh5  Kxf4
5.Kg6  Ke5
6.a5    !!      and black is forced to abandon it's pawn. 

I see now that the one square I should have avoided is g4 except when the black king is already on e4. But the point is white can force a way through & win from it.

Yes, Move 3 is the time for black to make his move.  3. Kf1 should be responded to with Kd7.

3. Kf1  Kd7

4. Ke2 or Kf2 leads to loss of zug. advantage for white when black answers with Kd8 or Ke8.  Ke1 is a repeat position, Kg1 is a rank 1 position and looks like black should have no problem recovering the advantage.  Kg2 looks to be best choice.  But if black stays on rank 8 and on same file or left of white, black can grab the advantage whenever white steps foot on rank 3.

4. Kg2  Kd8

5. Kg3  Ke8 

5. Kf2  Kd8

6. Ke3 Kc7

 hmmm, maybe 3. Kd2 can not be stopped.

But, I don't know that we have looked at 1. Kd1  Kc6 which immediately breaks the zug. 

If I were Black, I should abandon the f6 pawn, and go for the queenside pawns, where they would promote at the same time.

ChaoChinKun - It might be a good idea to abandon the f6 pawn but for one flaw:-

You get the idea ?  I can't find any square on which the black king can avoid the exchange of it's queen.  Then white simply walks it's remaining pawn up the board.

 Admittedly if you play 7...Kc4 instead of c3 it's still sticky for black but it can avoid the exchange.

I took a draw in a won position, then? Ugh.

Don't beat yourself up !  Look at the amount of time people have spent analysing your position & who is really certain that you could have forced a win even now ? 

I've gone from thinking this was a certain draw to maybe either-way, then a certain win & now I'm thinking a draw again given very best play by black.  I'm still not sure. And you were under time pressure !

Good luck with the chess.

With best play white wins.

My pink elephant certainly agrees on that.

I would say that, if it was a combination you were going to keep having to recalculate, that was 10 to 15 moves out,especially each time your opponent deviates from the original best line, it might be better to go ahead and draw.

If you opponent is a higher rated player, you stand to gain something. If your opponent is rated lower and then you try to play on and overlook one little thing, you really lose, if things go wrong. In a tourney, I could see you going the extra mile. This is the only way I bother, unless I really want to beat the person for personal reasons.

The last part of your statement is why I take a draw most likely.

1.Kd1 does not win, period.

It doesn't? Why doesn't it? I thought White could keep on placing Black in zugzwang, or did I miss something? Can you enlighten us?

After 3. Kf1 Kd7, both 4. Ke1 (zugzwang) and 4. Kg2! preserve the win.

After 4. Kg2:

• White answers 4...Ke7 with 5. Kf3, returning to zugzwang.
• White answers 4...Ke8 with 5. Kf2, returning to zugzwang.
• White answers any other king move with 5. Kg3, and black cannot stop white's progress along the h-file.

I'm still very confident that white wins after 1. Kd1. If someone wants to propose a move that saves black after 1. Kd1, I'll try to refute it (without looking at Houdini) to see if my method holds up. 

If an engine really evaluates the position as +10 then I will believe it. Have you actually checked? I would guess an engine would evaluate a position like this, assuming it is drawn, at between +1 and +2, maybe +1.39. If the engine really does evaluate +10, then it has probably calculated forced queening of a pawn.

3. Kf1 Kd7  4. Ke1  returns us to our original position which we then repeat 4. ... Ke7 5. Kf1  Kd7  etc.. so, yes white has zug. advantage but still stuck on rank one with this line.

as for 3. Kf1  Kd7 4. Kg2 .. on closer look, white does have zug. advantage, and black can not seem to break it.

So both  3. Kd1 Kd7  4. Kd2   and 3. Kd1 Kd7  4. Kg2 both appear to win for white. 

I'm kind of proud to have come up with Kd1-e1-f1 right away, obviously trying to break the opposition.  But like Pfren said it's just schematic thinking, this time turns out to give a win.

Seems others came up with the same idea... I guess there's not much else to try in this situation :)

Very sensible.