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I'm trying to study king & pawn endgames, where both sides have one or multiple pawns remaining. So far it seems really puzzling to me - while I know the basic principles of opposition and outflanking, I'm unable to apply them efficiently when there's more than one pawn remaining and I'm basically guessing every time.

I am trying to solve the following problem with white to move:

[FEN "8/p7/8/8/1K6/4k3/2P5/8 w - - 0 1"]

I can see no way how to do it efficiently, other than calculating move-by-move and almost certainly making a mistake on the way...

So, is there any simpler way, how to do this? I'm thinking about something like square-rule etc.

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    @MichaelWest It's white to move, I've also added it into the question now. Also, yeah, I can see it's really complex... So, if there's something I can hold on in the beginning, it'd be nice :)
    – Eenoku
    Commented May 25, 2023 at 16:51
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    Accurate calculation is the main skill a chess player needs. There are some general rules and principles but they amount to nothing if they're not backed up by lines that work.
    – David
    Commented May 25, 2023 at 21:12
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    While there are some rules and principles worth knowing about, nothing can replace concrete calculation in situations like these. In general, pawn endgames where the result is not completely obvious for some reason tend to be determined by one tempo in one crucial line, and it's just a matter of finding that specific crucial move in that specific crucial line to understand what is going on.
    – Scounged
    Commented May 26, 2023 at 0:24

2 Answers 2

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The first thing you need to know is the rules for K+P vs K, when is it a win, when is it a draw.

Apart from the square rule, which you already know, there are 3 rules.

1. For pawns on files b through g, when the pawn is behind the halfway line then the king has to be able to get to one of the 3 squares two squares in front of the pawn in order to win. Otherwise it is a draw:

[Title "b-g pawn behind halfway"]
[fen "8/8/3KKK2/8/4P3/8/8/8 w - - 0 1"]

2. For pawns on files b through g, when the pawn has crossed the halfway line then the king has to be able to get to one of the 3 squares one square in front of the pawn in order to win. Otherwise it is a draw:

[Title "b-g pawn crossed halfway"]
[fen "8/8/3KKK2/4P3/8/8/8/8 w - - 0 1"]

3. For a and h pawns the king has to get to b7 or b8 and g7 or g8 respectively in order to win. Otherwise it is a draw.

[Title "a and h pawns"]
[fen "1K4K1/1K4K1/8/8/P6P/8/8/8 w - - 0 1"]

In all of these positions, if black can stop white getting the king to one of the key squares marked then black can draw.

Looking at the position you give it is fairly straightforward to see that it is a win for white. If black didn't also have the a pawn then the plan would be to push the c pawn up the c file and the king up the b file as necessary to protect the pawn.

Since black does have a pawn, the one thing white has to be careful of is knowing what to do after black pushes the a pawn to a5, possibly with check and this requires a bit of calculation. (Note that if it were black to move first then this tactic would draw immediately as black can then stop white getting the king two squares ahead of the pawn.)

If this a pawn check comes after the moves 1. c4 Kd4 2. c5 then white must not capture because black would just take the c pawn. Instead white pushes the king on to b5 and although black can queen the a pawn, black queens the c pawn first and wins thanks to move Qh8+ skewering the black king on d4 against the black queen on a1.

[FEN "8/p7/8/8/1K6/4k3/2P5/8 w - - 0 1"]

1. c4 Kd4 2. c5 a5+ 3. Kb5 a4 (3...Kd5 4. c6 Kd6 5. Kb6 {Black can't stop the c pawn}) 4. c6 a3 5. c7 a2 6. c8=Q a1=Q 7. Qh8+

If black moves to e4 on the first move then white has time to capture the black pawn after a5+ and still come back to b5 to protect the pawn and shepherd it home.

[FEN "8/p7/8/8/1K6/4k3/2P5/8 w - - 0 1"]

1. c4 Ke4 2. c5 a5+ 3. Kxa5 Kd5 4. Kb6 {and with the pawn beyond the halway line white's king is on one of the key squares on square in front and queening is guaranteed.}

An excellent book which covers this and many other useful endgame ideas and principles is Amateur to IM by Jonathan Hawkins.

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Sorry to bring the news to you: Basically no, you have to calculate until kingdome come. This doesn't say the situation is hopeless, as there are a lot of basic principles beside those you already mentioned.

I made a pawn endgame primer for my chess kiddies, and here is a compressed version for you. You can simply google the keywords for more material.

Independent (not even restricted to pawns)

  • Calculate. If possible, count. (Square rule etc)
  • Find key features of the position
  • Make a plan
  • Make candidate moves
  • Calculate variants
  • Break if clear (usually: if promotion happens)

Dependent on situation

  • Square rule
  • Zugzwang (one- and both-sided), Tempo win and Tempo loss, Triangulation
  • Opposition (close, far, diagonal, general)
  • Bodycheck and Outflanking
  • Critical, mined and corresponding squares
  • Passers (simple, protected, far, races between)
  • Good (connected, side-by-side) and bad (lagging, double, isolated) pawn structures

Now to your position. It is quite hard (I'm an endgame god and I had to think :-) No wonder, it's an endgame study by the great Grigoriew, 1931. You can look into the other answer for details (typical engine warning: 2...Ke5 is just a feeble attempt to make White play 2.Kb5?? Kd5! with a two-side zugzwang. After 2.Ka5! Ke6 3.Ka6! -still no sell!- Kd5 3.Kb5 Black has only delayed the inevitable. Also now a6+ is not really the relevant line). These two moves you had to find, and they are easy to find once you see the zugzwang.

So, working down the lists:

  • Key features: White has the better king position and can guide the pawn as well as stop the enemy one.
  • Plan: Since Kc5/Kb5 are futile (wastes only time), plan is immediately pushing through the passer. Candidate obvious.
  • Calculating only after the sneaky Ke5. Also, in the double-zugzwang position, you must calculate 4...Ke6 5.Kc6 a5 6.Kb7 a4 7.c6 a3 8.c7 a2 9.c8Q+ or 4...Ke5 (now the same moves) 9.c8Q a1Q 10.Qh8+ (Black has a bad hair day).

So the motives are:

  • better king position
  • double zugzwang
  • mined field b5
  • passer race with two typical win motives: queening with check/skewer at the end.

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