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Another retrograde chess puzzle! Unless I overlooked some shortcut, this should be a bit more tricky than my last one:

In the position below, what was White's final move? Final position, FEN is 4k1R1/2ppppKR/pp4p1/8/8/1P4PP/B1PPPP1B/rb4br

In your answer, please provide your reasoning and a (sketch of a) proof game leading to the final position.

(Further) Edit: In regards to a shortest possible proof game, I can currently offer

Checkmate on White's 44th 42nd move.

Again, this feels quite optimal, but there may be some tricks to shave off one or two extra moves that I am missing here. If you find a better solution, please feel free to post that as an answer.

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3 Answers 3

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Proof game (GIF)

enter image description here

Preliminary analysis

  • First of all, White’s last move must be Rh8xg8 (captured on a light square), but we must determine what was captured.
  • (When I say a “real” piece I mean the piece does not come from the promotion of a pawn)
  • Because of b3, c2, f2, and g3, we conclude that one of White’s bishops is promoted, and thus all other White pieces are real.
  • Therefore one of the following two sequences must exist (the 2nd case is in brackets):
    • Promote the a pawn to a dark (or light) bishop
    • Place the promoted bishop in h2 (or a2)
    • Move g2-g3 (or b2-b3)
    • Get the f (or c) bishop out … (^)
    • Put that bishop in a2 (or h2)
    • Move b2-b3 (or g2-g3)
  • Since all other White pieces are real, both White rooks and White king must get out via the a and b files.
  • But somehow Black needs to insert pieces into White’s back rank during this sequence.
  • Note that (^) above is a blocker. In either case, before we reach (^), (at least) the White h1 rook will be stuck in this region and will interfere with black’s insertion of g1 and/or h1.
  • We now note that the 2nd case is impossible because White’s h1 rook is locked in too early. We now focus on the 1st (and only) case.

1st case is the only case

  • The Black dark g1 bishop is promoted (following 3 captures h->g->f->g) and never left its promotion square.
  • The c1 bishop (White’s dark bishop) is captured on its starting square, otherwise White’s h1 rook is locked in.
  • Therefore all of Black’s other pieces are real.
  • Statistics:
    • White is missing 4 items (1 dark bishop captured on starting square, 2 knights, 1 queen).
    • Black is missing 4 items (1 dark bishop, 2 knights, 1 queen)
  • There are complications with how the h2 bishop can get there after promoting on b8. Furthermore, since Black’s light bishop is real, its departure from its home square is complicated.

Sketch of sequence

  • The sequence must now be
    • Black pushes a7-a6; White takes axb. (In some order)
    • White’s a pawn arrives at b6.
    • White takes b6xa7 (captured on a dark square) and a7xb8=B (captured on a dark square to make a dark bishop). (White did 3 captures up to now — come back to which 3 pieces later)
    • White retreats via Bb8-a7 and then out again to one of c5, d4, or e3.
    • The following in some order
      • Black pushes b7-b6
      • Black gets out the light bishop
      • This sequence (has to be in order)
        • White pushes h2-h3
        • White places dark bishop with Bh2
        • White pushes g2-g3
        • White gets the f (light) bishop out
        • (Assume the white c dark bishop is already taken by a Black knight — come back to this later)
        • White gets its back rank out via a2
    • Black plays Rh1, then Ba2, then Bb1, then Ra1. Meanwhile Black also plays hxg, gxf, arriving at f3, then f3xg2, then g2-g1=B (dark).
    • White puts the Rooks and King into position. There are no tempo issues

Issues we ran into

  • Which 3 pieces did White’s a pawn capture? (And therefore which piece was captured on g8?)
    • The g8 capture is not the queen because of check issues. It is a light square so it is a knight.
      • This causes insertion complications but not unfixable.
    • So White’s a pawn captured 1 dark bishop, 1 knight, and 1 queen.
      • These can all get out via the g and h files before the sketch of the sequence above, possibly with the help of an early hxg (taking a White knight, for example).
  • Which 3 pieces did Black’s h pawn capture?
    • Recall the c1 dark bishop is captured on its starting square.
    • So the Black h pawn captured 2 knights and a queen. This poses no problems.

Proof game (PGN)

Feel free to optimise the proof game in your own answer.

1. Nf3 h6 2. Ng5 hxg5 3. Nc3 Rh4 4. Nb1 Nf6 5. Nc3 g6 6. Nb1 Bg7 7. Nc3 Nd5 8. Nb1 Nc6 9. Nc3 a6 10. Nb1 Bd4 11. Nc3 Ba7 12. Nb1 Kf8 13. Nc3 Kg7 14. Nb1 Qh8 15. Nc3 Kh6 16. Nb1 Qd4 17. Nc3 Bb8 18. Nb1 Qa7 19. a4 Nb6 20. a5 Ne5 21. axb6 Nc6 22. bxa7 Ne5 23. axb8=B Nc6 24. Ba7 Ne5 25. Bd4 b6 26. Nc3 Bb7 27. Nb1 Be4 28. h3 Nc6 29. Be5 Nd4 30. Bh2 Nc6 31. g3 Nd4 32. Bg2 Bc6 33. Bd5 Nb3 34. Nc3 Nxc1 35. Nb5 Nb3 36. Ra5 Nd4 37. Qa1 Nf5 38. Qa4 Nd4 39. Kd1 Nf5 40. Kc1 Nd4 41. Kb1 Nf5 42. Ka2 Nd4 43. Ka3 Nf5 44. Kb3 Nd6 45. Ra1 Rf4 46. Kc3 Rf3+ 47. Kd4 Rh8 48. Qb3 Rf6 49. Qf3 Rh7 50. Qf4 gxf4 51. Kd3 Rh8 52. Nd4 Rh7 53. Nf5+ Kh5 54. Ne3 f3 55. Ng2 Rh8 56. R1a4 Rf4 57. Rb4 Rh7 58. Rbb5 Ra4 59. Rc5 Ra1 60. Rcb5 Rh1 61. Rc5 fxg2 62. Be6+ Kh6 63. Rd5 Bb5+ 64. Kd4 Bc4 65. Ke5 Ba2 66. Kd4 Bb1 67. Ke5 g1=B 68. Rdc5 Kg7 69. Rd5 Kf8 70. Rdc5 Rh4 71. Rd5 Ra4 72. Rdc5 Ra1 73. Ba2 Ke8 74. Kf4 Kd8 75. Kg5 Kc8 76. Kh6 Kb7 77. Kg7 Ne4 78. Rd5 Nf6 79. Rh5 Ng8 80. Rh8 Kb8 81. b3 Kc8 82. Rah5 Kd8 83. R5h7 Ke8 84. Rxg8# 1-0

Comments

It’s a beautiful puzzle and I enjoyed solving it. Thank you for creating it. I wonder if I over-complicated my solution. At one point I panicked because I thought I overlooked some castling trick, but it turns out I didn’t.

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    $\begingroup$ Excellent analysis! This is (essentially, up to the dismissal of case 2) how I think about the solution as well, so no worries about over-complicating it - this is a rather intricate sequence of events to uncover :) Thank you also for the kind words at the end :) $\endgroup$
    – Tim Seifert
    Commented May 14 at 21:52
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Has to be Rook takes Ng8. It can't be the queen because of the obvious King Checking scenarios. Lichess Image

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  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented May 19 at 15:13
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    $\begingroup$ Thanks for the submission! This is a good first thought, but there is more to a correct solution I'm afraid. Why is it not a rook or a bishop that was taken, for starters? $\endgroup$
    – Tim Seifert
    Commented May 19 at 17:00
  • $\begingroup$ I may have rushed to a quick conclusion to begin. If your proposing that the Black Rooks and Bishops are Promotions then it is possible for one of them to be on g8. Unlikely a Rook because it would have already taken the White Rook on h8, otherwise it's Mate anyway. If Rf8-g8 Then Rh8xRg8#. Benjamin Wang also thinks it's a Knight but It could be a Bishop. I'm still figuring it out. $\endgroup$
    – J4ck B4ker
    Commented May 22 at 16:10
  • $\begingroup$ Well, I'm not proposing that they are, but it is a possibility to consider :) Note again that this is a strict logic puzzle, so that we can not assume the players to have any intention about the outcome at all! (As in, they do not care if they can avoid mate or win material or anything like that. The only "goal", if you will, is to reach this position using only legal moves.) Thanks for keeping at it and good luck figuring it out :) $\endgroup$
    – Tim Seifert
    Commented May 23 at 14:10
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It is:

Rxg8#.

It is pretty clear that

unless white moved the h8 rook, the black king would have hung itself last move.

Assuming that the game is valid,

white must have captured the g8 piece, since otherwise the black king would also have hung itself.

It's not possible to know which piece was captured:

Knight and queen are both possible.

To build a proof game:

Note that both pawns underpromote to bishops.

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    $\begingroup$ How can it be a Queen that was taken? The king can’t move into the G7 square in order for the queen to give it a check… $\endgroup$
    – PDT
    Commented May 14 at 13:19
  • $\begingroup$ Thanks for the submisson! The captured piece is in fact uniquely determined. But why are there only two options to consider? Some more details on the proof game might clear that up ... $\endgroup$
    – Tim Seifert
    Commented May 14 at 13:22
  • $\begingroup$ The other pieces are required for the pawns to achieve underpromotion. $\endgroup$
    – Sny
    Commented May 14 at 13:28
  • $\begingroup$ Not sure why this answer is so downvoted. It's technically correct except for not know which piece was taken $\endgroup$
    – Rumpelstiltskin
    Commented May 18 at 21:03
  • $\begingroup$ ^ plus which piece was taken doesnt influence what move it was. but it is true that i didnt build a proof game $\endgroup$
    – Sny
    Commented May 19 at 0:13

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