Pawn ending study - Nalimov

While looking for some long mates in Nalimov, came across this interesting position.

I couldnt believe White has only one move that wins, another that draws, everything else actually loses.

Can you find the correct continuation, (without peeking)?

[EDIT] The more I think about this, the more I am certain I would have lost this in a game : (

Gut feeling goes for 1.h5 and if Kxh2, follow with 2. g4, etc.???

Yes, h4 wins. g4 draws, but surprisingly trying to centralise the King by Kc6 or Kc7 actually lose.

I think h5 is most easily deduced through elimination of all the other moves (at least, that's how I did it). After all, Black threatens 1...Kxh2, where 2.g4 Kg3 and Black catches one pawn and then the other. If White plays 1.g4, 1...Kh3 and he still round up the pawns. That leaves 1.h5. Then there are two important lines to look at:

1...Kxh2 2.g4 Kg3 3.g5 Kg4 4.h6 and promotes

1...Kh3, which I couldn't figure out to the end in my head, but it looked winning since if Black ever took h5 then White could set up an h3-g3 pawn shield.

I just found an example of Nalimov's true insanity...

Wow, that's acually amazing.

I played throught that, what I thought was the most natural looking moves for White, and got mated with my knight still on the board.

After looking at some variations, I think it's a very elaborate Knight trap.

1. ... Nc2 is nice, sets up a trap.

2. Kxc2 is an immediate stalemate, while 2. Nxc2+ Kb1 wins the Bishop as it cannot move due to stalemate (again)

So 2. Nc6 is the only move that covers d4 (otherwise 2. ... Nd4+ and black has some breathing room). I guess that's the reason only 2. Nc6 wins... eventually.