pawn needs to get on g6 somehow?
White to Move and Win
#1
I would say the move is Kc3 bringing the king to the action. But after Kh7 by black I'm not sure what is the best move for white. Maybe Nf4 followed by Kh6 and Ne3?
It's a challenging endgame study, but I wouldn't go so far as to say that the moves are not intuitive. The initial 1. Kc3! clearly places black in zugzwang (a common endgame theme), and both of the difficult lines (1. ..Kg8 and 1. ..Kh7) can be counted out if you know your basic endgames and expect to have to sacrifice the knight to jailbreak the pawn (another common theme).
In both of these lines (any of the pawn moves just drop a pawn for nothing and lose trivially), black must weaken his position by moving. The first line 1. ..Kg8 sacrifices a critical tempo since in the immediate 1. Nf6+?? ( which was the first move I looked at here) white needs extra tempo to make this work, while on the second line 1. ..Kh7 2. Nf4! traps the black king on the h-file.
White to move and win.