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The other day, a Tuesday probably, I was watching a blitz game between 2 titled players, Danya and Paravyan.

They have about 1 minute remaining each, and from this moment

enter image description here

they both struggle to perform complex calculations within the narrow remaining time frame.

It would take amateur players (incl. myself) quite some time to try to accurately push the right pawn or move the king.

But I thought at that level this position must have already occurred, or, more likely, be somewhat close enough to a previous one, that the players would play in a more automatic fashion.

Obviously that's not the case ; even at the candidates, or other major tournaments in standard games, in a 3 vs 3 pawns endgame, players dedicate a huge amount of time between some of these final moves. This is seemingly still complex.

While mathematically, there is still a huge number of possible combinations, isn't a few pawns endgame - at that level - resemble a pattern the players know, and they could play more quickly, but they take time just to prevent a stupid mistake?

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    Naroditsky actually has a great video showing just how intricate and involved endgames can be with few pieces.
    – ajd138
    Sep 13 at 22:09
  • @ajd138 I don't have time to check right now but if I remember right, the 2nd video in that series (intro to pawn endgames) highlights a specific pawn endgame where everyone thought it was a draw, including the engine up to fairly high depth, but it's actually a win
    – llama
    Sep 15 at 13:17

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But I thought at that level this position must have already occurred, or, more likely, be somewhat close enough to a previous one, that the players would play in a more automatic fashion.

Not true. This isn't some position they've likely studied before because there's far too many pawns on the board for it to be a practical lesson in a book. If there's one or two pawns on the board, then yes it's entirely likely. But 5 for each side? No chance.

Both players clearly understand the principles at play here: get your king up in the endgame, try and create a passed pawn, don't allow your pawn majority to be frozen by a pawn minority, etc. However, king and pawn endgames are notoriously tricky. Seemingly similar moves can flip the evaluation completely on its head. Moreover, as opposed to other endgames, once you've moved a pawn you can never take back the move. So every pawn push is potentially a critical decision.

As you said, there's still a huge number of possible combinations, and so the position is not just seemingly still complex - it is still complex. If you have an extra minute on the clock and the game could turn on one brash decision, wouldn't you too use all your time to make sure you finish the game off correctly?

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But I thought at that level this position must have already occurred, or, more likely, be somewhat close enough to a previous one, that the players would play in a more automatic fashion.

Playing in an "automatic fashion" is a very good way to lose.

Let me illustrate with a trick question. Is the following position a draw or a win for white?

[title "Draw or white win?"]
[fen "4k3/8/8/8/8/8/4P3/4K3 w - - 0 1"]

Of course anybody who knows their king and pawn endgames knows that this is a draw if it is black to move and a white win if it is white to move. When such a simple position with the kings as far apart as they can be and just one pawn on the board depends on who has the move it is very clear that a position which is "somewhat close enough to a previous one" still requires very careful calculation. Playing on automatic pilot is a way to lose. Good players almost never do this in the endgame and when they do they regret it.

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    What is the trick in the question?
    – justhalf
    Sep 13 at 7:31
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    @justhalf That they don't say whose move it is.
    – Arthur
    Sep 13 at 7:40
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    Oh, right, the game in the OP. I think you missed the point that Brian Towers wanted to make. Not to put words in his mouth, but I don't think he literally meant to highlight "whose turn it is" as a factor in analysis. Rather, that in the above position, if it is white to move then white wins, but if you allow black to intervene with a single kong move first, it's a draw. The exact position of the black king, even with so few pieces left, even so far away from any white pieces, even with just 3 pieces on the board completely waarps the evaluation. The margins are likely the same in the OP.
    – Arthur
    Sep 13 at 13:39
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    I just had a brainfart, and will delete my comments. Your explanation makes sense, and it's also the way I understood Brian's point. Thanks. I was just confused by "the question".
    – Eric Duminil
    Sep 13 at 13:49
  • @EricDuminil I initially was confused as well, and asked something about the question, but shortly later realized what Brian meant and deleted my comments too, lol. I guess the example can be made clearer somehow, now that at least there are two readers confused. One point I was confused with was that the example is a puzzle, where we don't know whose turn it is, while the OP is an actual game, where whose turn is clear. The explanation link like Arthur described is a good addition to the answer, IMO.
    – justhalf
    Sep 15 at 7:51
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As others have mentioned, endgames in general are quite dependent on the specifics. It's true at some level that to grandmasters every endgame is "somewhat close enough to a previous one", but while these definitely inform the calculations, they're in no way a substitute.

To emphasize this point, even if a position is literally one pawn move away from something they've studied before, it could change the entire story. It's possible that all the calculation they did regarding the first position becomes totally irrelevant in the new one!

This is doubly true for endgames with only kings and pawns (as in the video you sent), which have a reputation of being extremely difficult to play perfectly. King and pawn endgames also much scarier for the players in the sense that they tend to be decisive (lead to a win for one player or the other, instead of a draw) compared to other endgames, so the stakes are higher.

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I cannot speak for the greatest, but it is probably still easier to calculate these positions than it is to memorize them; effectively selecting candidate moves is a learned skill, and subtle differences in position can potentially cause the engine to sway one way or another.

Now we can also think combinatorically: there are 56 squares which pawns can occupy. Suppose White has w pawns and Black has b pawns left on the board. There are now a total of 56P(w+b) different possible locations of White and Black pawns, and let's not forget about the position of the Kings! To put this into perspective, if each side had only one pawn left, there are about 1.5 million distinct positions, and this increases to over 90 million if one side has two pawns instead of one. It just seems absurd to put time into memorizing.

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  • Good point about memorization, but I am not convinced that 56P(w+b) is accurate. For one thing -- I think 56 should be 48 since once a pawn advances all the way then it ceases to be a pawn. For another thing using permutations doesn't quite make sense since e.g. you can't distinguish between a pawn on c3 and a pawn on d4 from a pawn on d4 and a pawn on c3.
    – John Coleman
    Sep 15 at 13:35
  • Some slight inaccuracy there (I think I must have meant 48) and should have estimated using multiple binomial coefficients instead of just the "lazy" way.
    – Andrew Chin
    Sep 15 at 17:30
  • I get the product of two binomial coefficients: C(48,w)*C(48-w,b), though this is maybe more of an upper bound since it isn't clear that all pawn configurations can arise from a legal game. For example -- can the white pawns and black pawns swap thier starting positions?
    – John Coleman
    Sep 15 at 23:30
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The difficulty arises because there are many important variations that must be calculated, which are strictly winning/losing/drawing within the margin of only one or two moves. Furthermore, though similar positions and themes often occur, the nuances in each position are unique. These subtle nuances can cause two "very similar positions" that are different only by a square or two to be winning/losing/drawing.

As an aside, Naroditsky discusses common themes in pawn endgames in this video. He mentions a couple of core concepts common to many pawn endgames:

  • Outside passed pawns
  • Protected passed pawns
  • Connected passed pawns
  • "Trousers"
  • Deep freeze

These (or combinations of them thereof) often play a role in various positions.

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