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I know that the queen is commonly said to be worth 9 points of material, but say we're in a queen vs. pawn endgame. Can a lone queen successfully defend against 9 pawns in any possible starting position? What is the limit for a queen defending against pawns? For that matter, what is the limit for each of the other pieces, assuming any possible starting position?

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    This strongly depends on positions of defending king and pawns. Even if white player has queen against single black pawn it may be drawn - that is if this pawn is on a2, h2, c2 or f2 and if black king is close enough to it, while white king is far away enough. See Wikipedia article on queen versus pawn. Very similar positions can be winning if black has one pawn extra, since in this case he has no stalemate tactics. Or if white king is close enough to force mate just after promotion. Jan 3 at 19:27
  • To wit, even 3qk3/8/8/PPPPPPPP/4K3/8/8/8 w - - 0 1 is drawish. Feel free to experiment with an engine and different positions. Jan 4 at 8:37
  • Very similar to chess.stackexchange.com/questions/33149/…
    – D M
    Jan 6 at 6:01

2 Answers 2

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If the side with pawns is to move, then it is quite simple to devise a position where just one pawn wins against the queen, by promoting and giving a checkmate.

If the side with the queen is to move, then the queen always at least draws against one pawn, but two pawns may win:

[FEN "K7/1PP5/8/8/8/k7/8/1q6 b - - 0 1"]
[StartFlipped "0"] 

1...Qe4 2. c8=Q Qa4+ 3. Kb8 Qe4 4. Qc7 Qe8+ 5. Ka7 Qa4+ 6. Kb6 Qb3+ 7. Ka6 Qd3+ 8. Ka7 Qe3+ 9. Qb6 Qe7 10. Qa5+ Kb3 11. Ka6 Qd6+ 12. Qb6+

The further the pawns from promotion, the more you need to win.

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It's almost impossible to answer this without knowing where the pawns, Queen and Kings are. That said, it normally would take a very large number of pawns to overcome a queen without any imminent threat of promotion. A graphic illustration of this would be "horde chess" on lichess.org, which pits an entire army against 36(!) pawns.

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