An 'easy' pawn ending

I've been coaching endings more frequently these days, due to a parent's request to work on her kid's endings. There was this particular pawn ending that I found intriguing as general principles of endings actually hinder the winning process and a strong and cautious player grounded in principles actually found it more difficult to make progress (compared to a younger and lower rated one).  As IM Kevin Goh once noted, "Some endings are all about pure calculation".

Now a small test for readers...

Black to play and win.

Now, what is the first move you thought of? Which direction would you shift your king to? Is your first instinct correct?

(The answer to this is provided in the analysis example below).

Here is the complete pawn ending problem I gave my students.

Hope you are still awake after this example!