5

We can ask a question:

  • What are the longest checkmates with the biggest material disadvantages? What, approximately, is the function between length of the checkmate and possible size of the material disadvantage? (I.e. what is the biggest possible disadvantage for a 10 move checkmate? 20 moves? 30? 40?)

If we ask that question, we'll mostly get answers where one side's extra material is hopelessly isolated. See "context examples".

So, the question I actually want to ask is more specific:

  • What are the longest checkmates with the biggest material disadvantages and positions which are as "open" as possible? (I.e. positions where one side's extra pieces aren't hopelessly isolated and aren't completely useless. See "example answers".) What, approximately, is the function between length of the checkmate and possible size of the material disadvantage?

Caveat 1: what matters is not only material disadvantage in the initial position, but the disadvantage throughout the checkmating variation. We're looking at material disadvantage at each move or even each half-move. We're asking "how long is the checkmating side down material?" and "how long is the checkmating side down at least N points of material?". If we have a position with checkmate in 100 moves, but the winning side is down material only the first 50 moves, then only the first 50 moves matter (or even less, if we're looking for specific N and the winning side is N points of material down less than 50 moves). Those rules are not set in stone, though. The main point is just that we care how material balance evolves throughout the variation. Material is counted with the standard value of pieces (Queen = 9, Rook = 5, Bishop or Knight = 3, Pawn = 1).

Caveat 2: illegal positions are allowed.

All positions below are created by me except where stated otherwise.

Context examples

[White ""]
[Black ""]
[FEN "brbqqqqq/rbrqqqqq/brbrqqqq/rbKbqqqq/brbrqqqq/rbrbqqqq/bPbrqqqq/kbrqqqqq w - - 0 1"]

1. bxc3 Kb2 2. cxb4 Kc3 3. bxa5 Kb2 4. axb6 Ka1 5. bxa7 Kb2 6. axb8=B Kc3 7. Ba7 Kb2 8. Bb6 Kc3 9. Ba5+ Kb2 10. Bb4 Ka1 11. Bc3#

In this position, White gives checkmate in 11 moves while never being less than 384 points of material down. Which means that at each move (even each half-move) of the variation White is down at least 384 points. But in the initial position White is down 416 points.

All credit goes to MonkeyBusiness and Cleveland. I've only insignificantly modified the position.

[White ""]
[Black ""]
[FEN "nRB4b/K1k5/pppppppn/qqqqqqrb/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq w - - 0 1"]

1. Rb7+ Kd8 2. Rh7 Bg7 3. Rxg7 Ke8 4. Rb7 Nf7 5. Kxa8 Nd8 6. Ra7 Nf7 7. Kb8 Nh6 8. Kc7 Kf7 9. Bd7 Kf8 10. Ra8+ Ke7 11. Rc8 Kf7 12. Kd8 Kf8 13. Be8 Kg7 14. Rc7+ Kf8 15. Rb7 Kg8 16. Ke7 Kg7 17. Bf7 Nxf7 (17... Kh7 18. Kf8 Kh8 19. Bg8 Nxg8 20. Ra7 Nh6 21. Rg7 Nf7 22. Rg8+ Kh7 23. Kxf7 Kh6 24. Rh8#) 18. Ke8 Kg8 19. Rxf7 Kh8 20. Rf8+ Kg7 21. Ke7 Kh6 22. Kf7 Kh7 23. Rc8 Kh6 24. Rh8#

In this position, White gives checkmate in 20+ moves while never being less than 352 points down. And White starts 358 points down.

We could also build upon the ideas of Boris Sidorov (1983, #22, 22 moves) and Otto Blathy (1922, win, 14 moves).

However, you can notice that in those positions extra material is completely isolated. In the last two examples the isolation is not absolute, but most of the extra pieces are completely useless (e.g. they can only be sacrificed in exactly the same way). What could we achieve in much more "open" positions where a bigger portion of the extra pieces are useful?

Example answers

[White ""]
[Black ""]
[FEN "qrqqqqqq/qqqqqBqN/qrrPnpqq/Nnr1nppr/qrBNPq1b/rP1B1p1k/pPQPQQRP/nNBNKR1b w - - 1 1"]

(I haven't included the solution in the diagram because you may want to experience it blind: https://lichess.org/study/hBcBtt33. If you want to reveal the next move - click "reveal the solution", it doesn't reveal the whole thing.)

This is a candidate position for a checkmate in 30+ moves while always being at least 126 points down throughout the entire variation. I verified the position in different ways (with Stockfish), but maybe there's still some super obscure line which computer missed.

Here's the link to immediately access the full solution and positions for verification and extention: https://lichess.org/study/v0lPLC2T.

[White ""]
[Black ""]
[FEN "b2b1Rqb/Nnp2brR/1pN1N1Pq/n2K4/pP6/2k1Pp1n/Rpp3br/rbBbrrqq w - - 0 1"]

1. Nb5+ Kd3 2. Ne5+ Ke2 3. Nc3+ Kf2 4. Nd3+ Kg3 5. Ne4+ Kg4 (5... Kh4?? 6. Rxh6+ Kg4 7. Ne5+ Kf5 8. Ng3+ Kf6 9. gxf7+ Rg6) 6. Ne5+ Kf5 (6... Kh5?? 7. Ng3+ Kh4 8. Nf5+ Kh5 9. Rxh6#) 7. Ng3+ Kf6 8. Ng4+ Ke7 9. Nf5+ Kd7 10. Ne5+ Kc8 11. Ne7+ Kb8 12. Nd7+ Ka7 13. Nc8+ Ka6 14. Nb8+ Kb5 15. Na7+ Kxb4 16. Na6+ Kc3 17. Nb5+ Kd3 18. Nb4+ Ke2 19. Nc3+ Kf2 20. Nd3+ Kg3 21. Ne4+ Kg4 22. Ne5+ Kf5 23. Ng3+ Kf6 24. Ng4+ Ke7 25. Nf5+ Kd7 26. Ne5+ Kc8 27. Ne7+ Kb8 28. Nd7+ Ka7 29. Nc8+ Ka6 30. Nb8+ Kb5 31. Rxb2+ Nb3 32. Na7+ Kb4 (32... Ka5?? 33. Bd2+ Nxd2 34. Rb5#) 33. Nbc6+ Kc3 34. Nb5+ Kd3 35. Nb4+ Ke2 36. Nc3+ Kf2 37. Nd3+ Kg3 38. Ne4+ Kg4 39. Ne5+ Kf5 40. Ng3+ Kf6 41. Ng4+ Ke7 42. Nf5+ Kd7 43. Ne5+ Kc8 44. Ne7+ Kb8 45. N5c6#

In this position, White gives checkmate in 45 moves while never being less than 67 points down. The position is a bastardized version of a composition by Otto Blathy and Heinrich Meyer. There White is only 17 points down.

It's potentially bad that all White's moves are checks though (most of Black's pieces are effectively frozen in place). So, here's another position:

[White ""]
[Black ""]
[FEN "rn2Qqrb/qq1nRb2/Nqp1nqkp/1q2PppR/RprpnQRp/brpPQrR1/qPPN2P1/rbBRK2R w - - 6 1"]

1. Rgxg5+ N4xg5 2. Qxg5+ Nxg5 3. Qxg5+ Qxg5 4. Rgxg5+ hxg5 5. Rxg5+ Kxg5 6. Nxf3+ Kh5 (6... Kg6?? 7. Nxh4+ Kh5 8. g4+ fxg4 9. Nf5+ Kg6 10. Rh6+ Kxf5 11. Rxf7+ Qxf7 12. Qxf7+ Nf6 13. Rxf6+ Kxe5 14. Bf4#) 7. Rxh4+ Kg6 8. Rh6+ Kg7 9. Rxf7+ Qxf7 10. Rh7+ Kxh7 11. Qxf7+ Rg7 12. Ng5+ Kh6 13. Ne4+ Kh7 (13... f4?? 14. Bxf4+ Kh7 15. Ng5+ Kh6 16. Qe6+ Nf6 17. Qxf6+ Kh5 18. Kf2!! Bxc2 19. Qf5!! Qxe5 20. g4+ Kh4 21. Nf3+ Kh3 22. Bxe5 Qf7 23. Ng1+ Kh4 24. Bg3#) 14. Qh5+ Kg8 15. Qe8+ Kh7 16. Kf2!! Rxg2+ 17. Kxg2 Nf8 18. Rh1+ Kg8 19. Rxh8+ Kxh8 20. Qxf8+ Kh7 21. Ng5+ Kg6 22. Qf6+ Kh5 23. Ne6!! Qg7+ 24. Nxg7+ Qxg7+ 25. Qxg7 Qd5+ (25... f4?? 26. Qf7+ Kg5 27. Qxf4+ Kg6 28. Qh6+ Kf7 29. Qh7+ Ke8 30. Qh5+ Kd8 31. Qh8+ Kd7 32. Qh3+ Ke7 33. Bg5+ Kf7 34. Qf5+ Kg7 35. Qf6+ Kg8 36. Qg6+ Kf8 37. Bh6+ Ke7 38. Qd6+ Kf7 39. Qf6+ Ke8 40. Qe6+ Kd8 41. Bg5#) 26. Kg3!! Qxe5+ 27. Qxe5 Kg6 28. Qe6+ Kg7 29. Qh6+ Kg8 (29... Kf7? 30. Qh7+ Ke6 31. Qg6+ Kd7 32. Qxf5+ Ke7 33. Bg5+ Kd6 34. Qf7!! c5 35. Bf4+ Kc6 36. Nc7!! Qa5 37. Qd5+ Kb6 38. Nxa8+ Kb5 39. Rxa5+ Kxa5 40. Qe6!? Bxb2 41. Qb6+ Ka4 42. Bxb8 Bxc2 43. Qa6#) 30. Qg6+ Kh8 (30... Kf8? 31. Bh6+ Ke7 32. Bg5+) 31. Bf4!! Qb5 32. Nc7!! Na6 (32... Rxa4?? 33. Qf6+ Kh7 34. Qe7+ Kg8 35. Bh6 Qe5+ 36. Qxe5 f4+ 37. Kxf4 Kf7 38. Qg7#) 33. Qh6+!? Kg8 34. Ne6!! Ra7 35. Qf8+ Kh7 36. Ng5+ Kg6 37. Rxa6 Bxb2 38. Qg8+ Kf6 39. Nh7+ Ke7 40. Bg5+ Kd6 41. Qd8+ Rd7 42. Be7+ Ke6 43. Ng5+ Ke5 44. Qh8+ Kd5 45. Qg8+ Ke5 46. Nf3#

In this position, White gives checkmate in 46 moves while never being less than 26 points down. Unlike in the previous example, here there's plenty of non-check moves.

It stands to reason that with 40+ checkmates we're approaching the absolute ceiling for large disadvantages in "open" positions.

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Stealing from myself, and presuming I fine tuned it correctly, here is a difference of 309-37=272 points.

[Title "me, chess.stackexchange.com 3/2/2024, Mate In 65"]
[FEN "qqqqqB1k/1qqqqB2/q1qqqp2/qq1qqNR1/qqq1qBnN/qqqr1RPR/qqqqqqPK/qRqRqR1b w - - 0 1"]

1. Rxg4 Qxg1+ 2. Rxg1 Qxg1+ 3. Rxg1 Qxg1+ 4. Rxg1 Qxg1+ 5. Kxg1 Qee1+ 6. Kh2 Qg1+ 7. Kxg1 Qdd1+ 8. Kh2 Qg1+ 9. Kxg1 Qcc1+ 10. Kh2 Qg1+ 11. Kxg1 Qbb1+ 12. Kh2 Qg1+ 13. Kxg1 Qaa1+ 14. Kh2 Qg1+ 15. Kxg1 Qaa1+ 16. Kh2 Qg1+ 17. Kxg1 Qaa1+ 18. Kh2 Qg1+ 19. Kxg1 Qaa1+ 20. Kh2 Qg1+ 21. Kxg1 Qaa1+ 22. Kh2 Qg1+ 23. Kxg1 Qaa1+ 24. Kh2 Qg1+ 25. Kxg1 Qb1+ 26. Kh2 Qg1+ 27. Kxg1 Qb1+ 28. Kh2 Qg1+ 29. Kxg1 Qb1+ 30. Kh2 Qg1+ 31. Kxg1 Qb1+ 32. Kh2 Qg1+ 33. Kxg1 Qb1+ 34. Kh2 Qg1+ 35. Kxg1 Qc1+ 36. Kh2 Qg1+ 37. Kxg1 Qc1+ 38. Kh2 Qg1+ 39. Kxg1 Qc1+ 40. Kh2 Qg1+ 41. Kxg1 Qc1+ 42. Kh2 Qg1+ 43. Kxg1 Qcc5+ 44. Kh2 Qg1+ 45. Kxg1 Qdd4+ 46. Kh2 Qg1+ 47. Kxg1 Qdd4+ 48. Kh2 Qg1+ 49. Kxg1 Qdd4+ 50. Kh2 Qg1+ 51. Kxg1 Qdd4+ 52. Kh2 Qg1+ 53. Kxg1 Qe1+ 54. Kh2 Qg1+ 55. Kxg1 Qe1+ 56. Kh2 Qg1+ 57. Kxg1 Qe1+ 58. Kh2 Qg1+ 59. Kxg1 Qe1+ 60. Kh2 Qg1+ 61. Kxg1 Qe1+ 62. Kh2 Qg1+ 63. Kxg1 Rd1+ 64. Kh2 Bxg2 65. Ng6#
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  • I think the structure of your position and its forced variation (in which Black just sacrifices everything on the same square) are more in line with the first category of positions, where extra material has no function. Edited the post to clarify that. But in any case, your position is great and your answer is very useful. It gives a possible upper limit for 50+ and 40+ checkmates. In your position, White is at least 85 (for 40 moves) or 40 (for 50 moves) or 4 (for 58 moves) points of material down. It may be impossible to top that by much, given the question's conditions.
    – Flowy Poosh
    Commented Mar 3 at 5:41
  • @FlowyPoosh Gotcha. What is math/logic behind those 85/40/4 though? Your question doesn't specify whatever function relation in have in mind regarding length ratio to initial points difference.
    – Rewan Demontay
    Commented Mar 3 at 8:41
  • Sorry, forgot to clarify that. Edited the post. Basically, I'm just looking at material disadvantage at each move (or even half-move). Your position is a checkmate in 65 moves, but White is not down material all those 65 moves. So, I'm looking at how much points White is down at each segment of your variation (first 30 moves, first 40 moves, first 50 moves). The exact function doesn't matter too much, what matters is that we care about measuring the disadvantage throughout the entire variation. Otherwise we'd get positions which simplify into long endgames.
    – Flowy Poosh
    Commented Mar 3 at 9:59
  • Played with your position for a bit. Seems like White's material could be reduced farther: qqqqqB1k/qqqqqp2/q1qqqp2/qq1qqpR1/qqq1qPnN/qqqq1PPR/qqqqq1PK/qRqRqR1b w - - 0 1 . If the engine doesn't accept such position, here's a slightly reduced version: qqqqqB1k/qqqqqp2/2qqqp2/3qqpR1/4qPnN/5PPR/qqqqq1PK/qRqRqR1b w - - 0 1
    – Flowy Poosh
    Commented Mar 5 at 9:17

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