How do you calculate pawn endgames, like if you have X number of pawns facing X numbers of opposite color?

Question

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.

Answer 1

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.

Answer 2

In most cases you don't mate with pawns only, so the plan in all pawn endgames is clearly to create a passed pawn and promote it. This already determines what you should look for in these positions, namely:

• passed pawns, and whether you can create one
• king activity
• weaknesses / easily attacked and capturable pawns

There are a few (somewhat related) concepts which become important in such endgames, such as:

• Corresponding squares: basically if your opponent moves the king to a certain square your king has to be moved to a specific square
• Mined squares: squares you should not go to with the king
• Zugzwang: since any player is forced to make a move, this can be used in certain positions when there is no good move available; forcing your opponent into a position to make a bad move
• Triangulation: a method to force a zugzwang position
• Opposition: one specific type of zugzwang
• basic king and single pawn versus king endgames: how to win or defend these; topics like: as attacker put your king in front of our pawn, occupying key squares, how to defend (aiming for opposition),...

Regarding your example diagram, I would evaluate the position as follows:

• black's king is more active
• slight imbalance of pawns (3 vs 2 and 3 vs 4, but because there are doubled pawns it will be difficult to create a passed pawn)
• the f4 pawn could be weak and if white protects it with g3, he has weakens the white squares on the kingside, particularly f3 giving black an attack route

It is obviously impossible to win the game on the kingside as neither side will be able to create a passed pawn there (even if white gives away the f4 pawn). So let's look at the queenside and how the pawn structure could change there. Basically there are four options:

1. black pawns on a5, b4, white on a4, b3, b2; This is a closed pawn structure, basically white wants to move the king via d3-c4-b5 to attack the pawns, while black wants to move his king via d3-c2 to attack the white pawns. Neither of this can be forced as black can just shuffle the king between c5 and d5 (covering c4) and white between d2 and e2 (covering d3). Neither player can make progress. Draw.
2. black pawn on b4, white pawns on b3, b2. This is similar to the first position ("1"), where the kings can be prevented from entering.
3. black pawns on a4, b5, white pawns on a3, b4, b2; This is the most interesting position. White wants to put the king on c2 and play b3 in order to create a passed pawn. Black wants to walk the king via c4 to b3. As a result when the black king moves to c4 white wants to move the king to c2 to prevent that walk; and likewise if the white king moves to c2, black wants to move the king to c4 to prevent b3. If there are no pawn moves available on the kingside (or none that lose there), this can result in a position of mutual zugzwang for whoever goes first to c4 (black)/c2 (white). 3a: white goes to c2 first, then the line could be 1. Kc2 Kc4, 2. Kc1 Kb3 3. Kb1 Kc4 4. Kc2 and white will play b3 and probably get a passed pawn and win; 3b black king goes to c4 first, then white king goes to c2 and creates a zugzwang position where black has to allow b3 afterwards
4. black pawn on a4, white pawns on a3 and b2. This is similar to the previous position ("3") with corresponding squares.

Taking this into account, it seems that black does not want to create position "3", i.e. black does not want to play a4 in the original position. If white is trying to achieve position "3" by playing b4 in the original position, black can capture on b4 after which white ends up with a weak pawn on b4 which will be captured.

As far as I can see, neither side can make any progress on the queenside.

This leaves as the only option for a win some combined play on king and queenside. A plan could be to place a pawn as far ahead as possible, (e.g. a black pawn on h3), then try to capture the blocking pawn (white pawn on h2 in this example) with the king. If you do this you give up the queenside, so need to calculate exactly who wins the race. However in this particular position I don't see how such plan could be forced and I believe it is just a draw.