How to figure out which squares are related?

The reason Ka2 is the only winning move has to do with a2 being a corresponding square to h8, meaning that if the white king goes to a2, black king has to go to h8, otherwise he will not be able to defend. This information is given but i have no idea how figure that out on my own!

My question is how do you figure out which squares are related like that? Is there a method/technique for doing it?

Thanks for your informative reply! I'll just imagine the board is a dance floor from the 70's and i'll do just fine!  

But seriously, great illustration & explanation - starting by figure out which squares my king has to reach in order to win and where my opponents king has to be in order to prevent me from reaching those squares, and then work my way until i see the path from those squares to both kings. I will start by practicing this method on simpler positions to get the hang of it.

Btw, consider making a blog post about this. This information is rare and i am sure many would like to learn how to navigate difficult kings+pawns endgames using the method you described!

Hahhahahahahahha the beginning of this reply made me laugh, seriously. Lol

Yes, very detailed information @bbmaxwell

I have successfully applied your technique on a simpler problem.

It is white to move and i wanted to find out where to start moving my king. I identified that if i can reach b4 square, i will win regardless of black king's position. From there i concluded that if i reach a2 square, black king has to reach b4 square to stop me, so i marked them with green. Each color represents related squares. It became obvious that if i play Kd1 then black will be able to draw. Playing Ke2 was the right move. Since black king always could access a red square if the white king steps on one, the correct procedure was to keep moving the king up and play d4 to access the black pawn.

Hopefully i will be able to do this kind of analysis in my head during an actual game next time

For the position in #8 that works well when it's black to play and determining which move draws (which is what it's intended to be in the original Grigoriev study) but in terms of white to play, the winning idea of playing d4 and Kd3 is so obvious that only Ke2 or Kf2 makes any sense at all, and over-analysing it in laborious detail doesn't make sense and is actually just unnecessarily complicating something extremely simple.