The distant opposition game.
White to play and force his King to either f8 or h8
Armed with the knowledge in this post, you will be able to do it every time, and stop everyone who does not know opposition theory from doing it to you.
First, what is distant opposition
It's when we can see that the Kings from far away will end up in a direct opposition situation like those discussed above.
How do you gain the advantage of distant opposition?
Method #1 - the simplest: leave an odd number of squares between the Kings, while the Kings are on the same file, rank, or diagonal.
Method #2 - indirect: move your King such that the Kings form a rectangle with squares of all the same color
OK, once we have the opposition, what do we do with it?
Establish a direct opposition, then OUTFLANK the enemy K to reach your goal.
Let's see this in action from the first diagram.
The natural move is 1.Kb2? but it fails against correct play.
So what about that rectangle thingy?
Same goal, different start position.
The easy way to remember all of this is SAME COLOR = GOOD - orthogonally, it always means you have an odd number. Every other case (even the diagonal) is solved by making rectangles with same-color corners. (This is extremely useful for blitz games where you don't have time to recall all that blah blah blah above.)