Some simple-but-important pawn endgames

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Cupine
Just randomly thought to create this thread. These endgames are not complicated but still easy to mess up if you don't know what you're doing. Might be useful for someone.
Let's start with this one. White to move and win.

 

Cupine
This one is more tricky but pretty neat. Also, the puzzle editor is horrendous, geez.
Anyway, white to move and win

 

Cupine
An initially counterintuitive but highly useful lesson in board geometry. If you understand the principle, it'll save you a lot of points. White to move and... I'm not telling, figure it out.

 

Arisktotle

Advanced level pawn endings! 

If you fill in the "Result box" in the "Details" screen of the puzzle editor you can replace the 1/2-1/2 with 1-0.

Cupine

Yeah, thanks, I did it, except for the third one, to avoid spoilers.

Cupine
A similar one. If you cracked the one in #3, this should be easy.

 

Alena_str
Thanks for posting it. They are really very useful.
Alena_str
The second puzzle is hard to understand quickly.
Jenium

In #3: Why can't the White king just move to c7 and grab the pawn?

Cupine

Because black grabs white pawn first and then white king is in no position to stop the promotion.

Jenium

Oh I see, Black has Ka6 in the end... Thanks.