How do you calculate pawn endgames, like if you have X number of pawns facing X numbers of opposite color? I'll use a random example here with white to move, but I'm really asking about the general principles. 

8/7p/6p1/pp3p2/3k1P2/PP6/1P1K1PPP/8 w - - 0 1 


I'm a chess amateur and endgames drive me crazy.  Every pawn can move in limited ways but when there are several, suddenly there are so many variations based on which one you move first and how the enemy reacts.  Do chess experts really calculate all that?  Or do they look and see, for instance, three pawns facing three pawns and right away know that's draw if king is located in a certain square?

In short, I'm sort of looking for some rules of thumbs that can help me analyze (like that pawn square rule about catching a pawn before it gets promoted) just to make pawn endgames manageable.
Do chess experts really calculate all that?


No, they really don't.

In a position like the one you gave an expert would look at two things -

1) The opposition

2) Reserve moves

and only then start calculating.

The opposition is what you get when the kings are opposite each other with one square gap between them exactly as shown in your diagram. The player who "has the opposition" (a good thing) is the player who doesn't have to move. 

The reason it's a good thing is because currently neither king can move forward because then the kings would be next to each other. Generally speaking the player who has to move their king first would have to move either sideways or backwards letting the other king advance when maybe they can attack the other sides pawns. 

Here it's not so straightforward because white pawns stop the black king from going to c3, c4, e3 and e5 and black pawns stop the white king from going to c4 and e4. White might want to play f3 to cover e4 instead of e3 and black might want to play b4 to cover c3 instead of c4.

Reserve moves are all the moves each player can play before he has to move his king and lose the opposition. The more the better and one reason to be careful how you push your pawns in an endgame. Here white has 4 obvious reserve moves - f3, g3, h3 and h4. Black has only 2 - h6 and h5. If white plays his moves in the right order, f3, g3, h3 first (in any order) then he will get extra moves g4 and g5 if he wants them because black cannot take the g4 pawn without giving white a passed pawn. So right from the get go black is in bad shape. He is going to have to give way and let white advance.

So, black's only hope is that he can construct a fortress with his pawns so that even though he will have to let the white king come to d3 he can stop it getting to the fourth rank using his king and pawns. 

Now it's time to do some calculation. 

It looks at first glance that maybe black is just going to be able to go back to d5 with his king (once his reserve moves are used up) and then shuffle his king between d5 and c5 because that will stop the white king going to c4 or d4 and the f5 pawn stops king to e4.

The problem is that then white is going to play b4 when his king is on c3 and black's king is on d5. This will give white a passed pawn and the win after either black takes the pawn and white recaptures with the king or black pushes the pawn to a4 and black plays b3.

So, maybe black needs to play b4 himself first? Maybe on his first move? Now it doesn't matter what white does. If he takes on b4 then black recaptures with his a pawn and his pawn is on b4. If white pushes a4 then black's pawn is protected on b4 and if white leaves it then so does black and the white king is never going to get to c4 to double attack it because the black king can always shuffle between c5 and d5.

But!!! It is white to go first. If he wants to win he has to play b4 himself on the first move. Otherwise it's going to be too late. So, what happens next? Black must capture with the a pawn and white moves Kc2 forcing black to take the a3 pawn as well. White recaptures with his a pawn and he has broadened the base and has the distant pawn (which means that after they take each other's pawns white is going to get back to the kingside just before white). 

An expert white would probably stop calculating here confident that he would win.

Let's look at how it might go.

8/7p/6p1/pp3p2/3k1P2/PP6/1P1K1PPP/8 w - - 0 1

1. b4 axb4 2. Kc2 bxa3 3. bxa3 Kc4 4. h3 h6 5. g3 h5 6. f3 Kc5 7. Kc3 Kc6 8. Kd4 Kb6 9. g4 Ka5 10. Kc5 Ka4 11. Kb6 Kxa3 12. Kxb5 Kb3 13. gxf5 gxf5 14. Kc5 Kc3 15. Kd5 Kd3 16. Ke5 Ke3 17. Kxf5 Kxf3 18. Kg5 Kg3 19. h4


and the f pawn queens.