That's a really in depth intro to pawn endgames! Nice
Pawn Endgames: Simple Pawn Endgames

(Alright, I am back now)
Creating Passed Pawns
All pawn endgames are decided to their result based on when or if a passed pawn appears. When you are up a pawn (which is not doubled) in a pawn endgame, you are often guaranteed a win due to the ability to make passed pawns so easily. But how do you do so? Well, again there are three different types of passed pawns you can create, but the general way to do this is to trade off pawns at a place where you have more pawns on that side of the board than your opponent does:
Distant Passed pawns
These passed pawns are just those (as shown in the name) that are way too far away for the King to catch it. These are the easiest passed pawns to win with as often, pushing the pawn if the King tries to catch it will prove that it is not possible for the King to catch it (if you follow the square of the King rule). Below is an example of a distant passed pawn, the King is too far away to catch it (and it shows again why square of the pawn rule is so useful):
Distraction Passed pawns
These pawns can be caught by the King, but when they are caught, the King will be distracted away and you can take other pawns, or at the very least infiltrate with the King before you can do so:
Protected passed pawns
These pawns do not actually ever need to advance! Their existence just decides the fate of the King as they cannot be captured due to protecting pawn, and if the King ever take the protector, this passed pawn becomes a distant passed pawn:
Of course, one thing that is very important about passed pawns is assisting them with your King (unless they are distant). Distraction passed pawns also need to be assisted for some time before sacrificed, as the sacrifice becomes useless if the infiltration is not effective.
Pawn endgames are the most simplified endgames to the point that you can directly calculate it all the way to the end result. They are simple without many pieces on the board, but are rather in fact very complex as you will still have to be very accurate when calculating pawn endgames. However, there are still some ideas that you can understand to help with your calculation:
The rules mentioned above can still be broken if you are able to get a better win or draw in a line, but YOU MUST CALCULATE IT. Again, do not underestimate the complexity of a pawn endgame, but in this lesson we will be looking at how to win pawn endgames up a pawn to make it simple (all you need is an extra pawn without any compensation to win the game). We shall now look at the following rules mentioned above:
Seizing the opposition
Basic Opposition
In King and pawn endgames, the Kings become the most powerful (and fighting) pieces on the board but cannot checkmate by themselves for obvious reasons. But the biggest difference between wins and losses often appear in something called the opposition. The Kings cannot attack each other directly, but the opposition is a magical property which allows either one of the Kings to straight up force the other away or out of the way through a sort of staring contest:
The first K to move does not have the opposition and the other K can chase the other K with this. With no pawns on the board, this is obviously drawn, and difficult to demonstrate the point, but this is how the typical opposition looks like. Let us now see what it looks like in action:
Okay, that's good and all, but why did white push the pawn at the very end and not earlier? Let us take the same position again:
As you can see, by giving black the opposition for just one move, they were able to force the draw. This itself shows that it is still possible to mess up pawn endgames with such simple and logical looking moves.
Other Oppositions
Now that we have seen what the basic opposition looks like, we will now look at other (more complicated) types of oppositions. We will first look at this position below:
It is white to move. Who has the opposition? From a little calculation, it becomes clear that white does. Perhaps you may have calculated that when the Kings meet in the middle, white has the opposition:
But what if black takes a different route? For instance, let's look at it again, but this time Black goes for a trickier route:
As you can see, it is much more trickier, but along with the long distance opposition (which was shown as to how this was an opposition in the first line show in this position if they directly went in to the centre). But how exactly was this game won?
Let's take a look at this position below: