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Suppose the end game contains three knights and king only for white, and one queen and king only for black. Which side is considered to be in a stronger position?

Value-wise, the knights are worth 9 points, while the queen is worth 8, so I would have considered white with three knights winning. However, I played this against the computer, and found my knights very ineffective, so I’m not sure.

If the answer depends on who is to play and the position of the pieces, that can be stated, but let’s not assume the pieces start in a highly specific configuration.

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  • 3
    You might be interested: chess.stackexchange.com/questions/30135/…
    – Allure
    Jan 11 at 2:46
  • It really depend how you would play it. Queens have much stronger movements, while knights do not.
    – user26887
    Jan 11 at 3:13
  • 3
    It seems that it trivial to achieve a draw for side with the queen, just exchange the queen for one of the knights and it should be a draw in most positions. Therefore, I think the side with the queen should not be worse.
    – Akavall
    Jan 11 at 4:25
  • Anyway, I hope that someone, who promotes a pawn to a knight, loses
    – d4zed
    Jan 11 at 8:16
  • 1
    "while the queen is worth 8" citation needed Jan 11 at 14:17

1 Answer 1

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The queen is not usually assigned a value of 8 pawns, rather often between 9 and 10 pawns. Of course the question still stands, assuming this roughly equal material balance who has the edge?

Let's consider two scenarios how the position could simplify: One is the queen sacrifices itself against a knight, or gets won by a knight fork for a knight. In this case one side stays with two knights against just the king. With no other pieces this is an easy draw for the lone king.

The other scenario is the queen wins a knight. In this case there exists a fortress for the knights like so https://lichess.org/analysis/4k3/8/8/3NN3/4K3/8/8/q7_w_-_-_0_1 (e.g. Ke4 Ne5 Nd5 against a king on e8). The idea is the two knights block the opponent king from approaching. (as a funny aside, since both positions are a draw, in some sense blundering the queen for free in this position is not a mistake, it's a draw before and afterwards) It is worth noting that this position might not always be reachable for the knight side, so if the pieces start out in unfortunate positions it may not be possible to reach this setup. Yet, with an extra knight the vast majority of positions will be a draw.

To conclude, this position is very drawish, neither side has almost any winning chances. In blitz both sides may have slight winning chances; the queen side might win a knight while keeping the opponent away from the fortress position, conversely it might blunder the queen to a knight fork without getting a knight back. However, overall I would still see the chances as roughly equal.

It might be interesting to look at adding more material for both sides. I suspect if you give both sides a pawn or two, the side with the three knights might have the advantage as knights are quite efficient at blocking checks, and their sheer number might overload the other side from defending their pawns sufficiently often, all the while pushing the own pawn. But without pawns, the knights are simply not enough to win.

Addendum: Since it was requested, here are the stats from the tablebase, note that those are very misleading since many positions are unlikely to be reached in practice: With the knights to move, (1) draws, (2) wins for the knights, (3) wins for the Queen, (4) would be wins for the Queen but prevented by the 50 moves rule.

(1): 2704444173

(2): 1164961530

(3): 796423773

(4): 746382

With the queen to move:

(1): 1478314845

(2): 42978066

(3): 2554099491

(4): 1078938

As can be seen by the huge difference between the knights to move and the queen to move, the vast majority of these are decided by tactics straight away, which is why being to move is a huge advantage in a random position. In practice however, you wouldn't really consider them as static endgames but rather trading down into a smaller endgame straight away.

If you want to argue that way you can see that the queen indeed wins more often, likely because 3 knights have a lot of options to be very misplaced on the board.

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  • It would be interesting to simply count the number of all drawn, lost and won positions by a tablebase (the data surely exists). I guess the number of forced forks is far lower than the number of the queen picking up two knights. Jan 12 at 13:17
  • @HaukeReddmann added the stats, but these stats usually don't paint an accurate picture. As can be seen by the fact that here overall less than 50% are drawn, which clearly is not what you'd see in practice reaching those positions naturally. (well, if you naturally get 3 knights against a queen...)
    – koedem
    Jan 12 at 15:13
  • To give an even clearer example of that fact, more than a quarter of the KR vs KR positions are not a draw. And clearly this is about the most drawish endgame that you could have.
    – koedem
    Jan 12 at 15:14
  • @koedem In some of those, the turn player captures the rook immediately or wins it using a skewer check. If such positions are excluded (as you'd only classify the endgame once it reaches a quiet position) I wonder what percentage are draws.
    – Rosie F
    Jan 12 at 17:21
  • @RosieF well yes, exactly. That is the point I'm making, these numbers are not very representative. I suppose one could look at positions where the next capture is at least 3 moves away or so?
    – koedem
    Jan 12 at 17:26

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