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I originally wanted to post a new retrograde chess puzzle today. But unfortunately, before taking the picture of the final position, I accidentally knocked some of the pieces off the board :( All I know is that one of them was supposed to be standing on f1 for some reason, but that's about all I can remember. Can you help me out here?

Which pieces fell off and where were they standing?

Final position

(14+14+?), FEN: Qq2K1rn/1Rpppp1r/k5pR/pp5p/P6P/1P4P1/1NPPPP2/n6N

Please provide your reasoning in your answer. Have fun solving this! :^)

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    $\begingroup$ As usual they arrived at this position through a sequence of legal plays starting from the standard opening position? That looks tricky to achieve as currently both kinds are in check but then it is a puzzle :-) $\endgroup$
    – quarague
    Commented Apr 10 at 9:50
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    $\begingroup$ @quarague Yes, the (completed) position was achieved in a legal game. So somehow, these checks need to be adressed :) $\endgroup$
    – Tim Seifert
    Commented Apr 10 at 10:08

2 Answers 2

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What a tricky puzzle! Solving it required a leap of faith.

All 4 missing bishops fell off. The white ones were on f1 and f8; the black ones, on a7 and c8.

I will not hide the explanation.

Start by realizing that a bishop must be on c8, otherwise bQb8 would be delivering an impossible check.

Similarly, we can assume a bishop is currently on a7, blocking the wQa8 check. This time there is a minor alternative: a white bishop may have just discovered check from a7, but if that is the case, let us rewind one move and focus on the position where the bishop is on a7. (We will soon see that the a7 bishop must be black, ruling out this corner case.)

Now, let us focus on the top-left corner.

Top left corner

What was the last piece to move in this cluster of pieces?

  • The pieces on a8, b8, c8 and a5 are stuck (note that pawns did not capture in this game, since that would require at least 2 captures but, accounting for the existing bishop on f1, at most 1 piece has left the board).
  • The rook cannot have moved last, because it would have been checking from b6.
  • The king neither, since on b6 he would have been receiving a check by a rook that is stuck.
  • How about the pawn on b5? Nope, since retracting it to b6 would leave a cluster of pieces (on b6, a7, b7, c7, d7, a8, b8, c8) that is completely stuck.
  • Finally, how about the a7 bishop? Could it have performed the last move? Well, maybe, but would not that require that their last move captured another bishop on a7? (To deal with the seemingly impossible wQa8 check.) Hum, that is not helping. If another bishop reappears on a7, we get the same cluster to resolve.

At this point I almost gave up. It seems the position must be illegal. Was not our argument exhaustive? I admit it was my faith in the composer the only thing that kept me trying, revisiting the logic again and again.

There is a minor detail I have not mentioned yet. If the bishop on a7 is black, we can retract it without making another bishop reappear on a7. I had discarded this case too, because it seems that position cannot be retracted further. But I was wrong! Can you see why?

Yes! One of the white pieces that is currently busy somewhere else may have visited this corner of the board at some point, to deliver a discovered check. Indeed, a white knight could have moved from a7 to c6 right before our black bishop made their last move to a7.

This resolves the most part of the mystery. The rest can be summarized as follows.

In order for Black to have enough retractions in the puzzle position we need a bishop on f8 to block the bRg8 check. This bishop on f8 has to be white, since the black dark-squared bishop rests on a7.

Finally, what is the color of the bishops on c8 and f1? It turns out the bishop on f1 must be white in order for White to have freedom of movement moments before the top-left corner cluster is finalized.

This leaves us with a black bishop on c8, completing the mystery.

I conclude with a proof game.

  1. h4 a5 2. Rh3 Ra6 3. Rb3 h5 4. Nh3 Rah6 5. Rb6 R6h7 6. Ra6 Nh6 7. Ra7 Rg8 8. Ng5 Ng4 9. Ne4 Ne5 10. Ng3 Ng6 11. Nh1 Nh8 12. g3 b6 13. Bh3 Ba6 14. Kf1 Bd3 15. Kg2 Na6 16. Kf3 Bg6 17. Ke3 Nc5 18. Kd4 Ne6+ 19. Kc4 Qa8 20. Qg1 Kd8 21. Nc3 Kc8 22. Qg2 Qf3 23. Ra8+ Kb7 24. Rb8+ Kc6 25. Nd5 Kd6 26. Kb5 Nd4+ 27. Ka6 Nf5 28. Kb7 Kc5 29. Kc8 Nd6+ 30. Kd8 Kb5 31. Nb4 Qa8 32. Qb7 Be4 33. Qa7 Bb7 34. Bf1 Bc8 35. Rb7 Qb8 36. Qa8 Ne8 37. Nc6 Rh6 38. Na7+ Ka6 39. a4 Rf6 40. Ra3 Rg6 41. Rf3 Rh6 42. Rf6 Rh7 43. Rh6 g6 44. Bg2 Bg7 45. Bf1 Bd4 46. Bg2 Rgg7 47. Bf1 b5 48. Nc6+ Ba7 49. Nd4 Nf6 50. Nf5 Ne4 51. Nd6 Nc5 52. Ne8 Nb3 53. Bg2 Na1 54. b3 Rg8 55. Bb2 Rf8 56. Bg7 Rg8 57. Bf8 Rgg7 58. Nd6 Rg8 59. Nc4 Rgg7 60. Nb2 Rg8 61. Bf1 Rgg7 62. Ke8 Rg8

Thanks again for this amazing composition!

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    $\begingroup$ Very nice work Miguel, you nailed it! :) The disentangling of the top left quadrant and the tempo issue in the middle were exactly the two critical observations I had in mind, so good job figuring them out :) $\endgroup$
    – Tim Seifert
    Commented Apr 13 at 12:41
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    $\begingroup$ Oh, and that graphic with the ambiguous bishops is really cool! Can I ask how you made that? $\endgroup$
    – Tim Seifert
    Commented Apr 13 at 12:44
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    $\begingroup$ I made it with GIMP: gimp.org $\endgroup$
    – Miguel Ambrona
    Commented Apr 13 at 14:45
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    $\begingroup$ @TimSeifert I have just made this: chasolver.org/viewer. The syntax is straightforward, e.g. try pasting the following FEN: Qq(Bb)1K1rn/(Bb)Rpppp1r/k5pR/pp5p/P6P/1P4P1/1NPPPP2/n6N $\endgroup$
    – Miguel Ambrona
    Commented Apr 13 at 15:58
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    $\begingroup$ Oh nice, that's impressive! $\endgroup$
    – Tim Seifert
    Commented Apr 13 at 16:53
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- No pawns are missing, so there have been no promotions - The only pieces not accounted for are the four (4) bishops - None of the white pieces moved last: the pawns and knights could not have revealed check; one rook is trapped; the other rook could only have moved from a7 or b6; the queen could only have moved from a7; the king would have had to move from check into... check - The only candidate for which piece moved last is the rook at g8

This means that:

1. The white king wasn't already in check. Therefore a bishop must be between it and the black queen (occupying c8 or d8). There's no way for a bishop to enter d8, so there is a bishop at c8. 2. With c8 and f1 both being white spaces occupied by bishops, there are at most two bishops on black spaces. 3. The king at a6 is not in check, so there is a bishop at a7.

EDIT: After realizing what the final move was, I forgot to use the fact that the king is actually in checkmate! This means that any adjacent (or orthogonal) space is defended by black:

- d8 and f8 are defended by the rook at g8 - f7 is defended by the rook at h7 - d7 is defended by the bishop at c8 (which must be black --> f1 is a white bishop) - e7 must be defended, by a black bishop (which means a7 is a white bishop)

So this black bishop could be at d6 or f6.

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    $\begingroup$ Thanks for your answer! These are some good thoughts, but as of now, there are still quite a few aspects of the puzzle that are left unadressed (for instance, why couldn't the last move have been made by a missing piece? What about discovered checks? What are the colours of the pieces you believe to have identified?). Answering all these may require a little more digging :) $\endgroup$
    – Tim Seifert
    Commented Apr 11 at 8:23
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    $\begingroup$ @TimSeifert - thanks for the prompt. I neglected a key piece of information! $\endgroup$
    – The Chaz 2.0
    Commented Apr 11 at 15:12
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    $\begingroup$ Also I don't know how to format within spoiler tags $\endgroup$
    – The Chaz 2.0
    Commented Apr 11 at 15:13
  • $\begingroup$ Hm, I don't really understand your reasoning now. The position is not necessarily checkmate, at least that is not a given information. To locate the missing pieces for certain, it is necessary to look a bit further into the past $\endgroup$
    – Tim Seifert
    Commented Apr 12 at 10:23
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    $\begingroup$ I see, maybe I could have worded that a bit better. (But note that a game does not necessarily have to end with checkmate.) To see why your solution cannot work, I would recommend checking out Miguel's accepted answer. In short: The top left cannot be untangled if we have a white bishop on a7 and all other pieces are still on the board (otherwise, what was the last move in that corner?) $\endgroup$
    – Tim Seifert
    Commented Apr 14 at 15:25

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