Endgame H and G pawns vs H pawn

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MidGameWizard

Hello, I have trouble drawing with White consistently in this position. I try to get close oppostion but he has g6 at his disposal to waste a tempo. Does anyone know the reasoning behind this to draw as white ?

Toldsted

I think you don't take the opposition, and when Black does then you have h5.

tygxc

#1
This is an exercise in conjugated squares.
Only 1 Kg3 draws. 1...Kf7 2 Kg4 2...Kf6 3 Kh5