Can you solve the fates of the missing chess pieces?

Question

Here is another retrograde puzzle from me - this time getting by without any underpromoted pawns if you can believe it! :) In the game leading to the position below, both players have lost three of their pieces. While the fates of the two light squared bishops are evidently not that compelling, the other four missing units tell a more interesting story:

Can you solve the fates of the two white knights, the missing black rook and the black queen? That is, can you determine the relative order that those pieces were taken in and, for each of them, find out where they were taken and by which piece?

In your answer, please provide your reasoning as well as a proof game (or a sketch thereof) leading to the final position.

Answer 1

Here is a proof game sorry for the messiness:

Chess analysis board
1. Nf3 Nc6 2. a3 Ne5 3. Nd4 e6 4. f3 Ng6 5. Nb5 Ke7 6. Nd6 Nf6 7. Nxc8+ Ke8 8. Nd6+ Ke7 9. Nc3 Ng4 10. Nde4 Ke8 11. Nd5 Qg5 12. Nb6 axb6 13. Ng3 Ra4 14. Ne4 Rd4 15. Ng3 Rd3 16. cxd3 Ne3 17. Ne4 Nxf1 18. Ng3 Ne3 19. Kf2 Nf5 20. Ne4 Nd6 21. Ng3 Qf6 22. Ke3 Nb5 23. Qf1 Nd6 24. Qf2 Nb5 25. Rd1 Nd6 26. Qg1 Nb5 27. Qh1 Nd6 28. Rg1 Qh4 29. Nh5 Ke7 30. Ng3 Kf6 31. Nh5+ Ke5 32. Ng3 Kd5 33. Nh5 Kc5 34. Ng3 Kb5 35. Nh5 Ka4 36. Ng3 Kb3 37. Nh5 Kc2 38. Ng3 Nf4 39. Nf1 Kd1 40. Kd4 Ke1 41. Kc3 Kf2 42. Ng3 Qg5 43. Kb3 Qc5 44. Rd1 Ne4 45. Nf5 Ng5 46. Nd4 Ngh3 47. Nc2 Ng1 48. Ne1 Kf1 49. Ka2 Qf2 50. Nc2+ Qe1 51. Nd4 Kf2 52. Nf5 Ngh3 53. Nh4 Qg1 54. Ng6 Ng5 55. Ne5 Nd5 56. Ng6 Ne4 57. Nf4 Nb4+ 58. Kb3 Nc2 59. Ka2 Ne1 60. Nh3+ Kf1 61. Nxg1 Bc5 62. Nh3+ Bg1 63. Kb3 Nd6 64. Nf4 Kf2 65. Kc3 Nc2 66. Nd5 Nd4 67. Ne7 N4f5 68. Nc6 Nh4 69. Kd4 Ng6 70. Kc3 Rg8 71. Kd4 Nh8 72. Ke5 g6 73. Kf6 Ne8+ 74. Ke7 Nf6 75. Ne5 Ne8 76. Kd8 Rf8 77. Kc8 Rg8 78. Kb8 Rf8 79. Ka8 Rg8 80. Nc6 Rf8 81. Nb8 Nd6 82. Ka7 Rc8 83. Nc6 Nc4 84. Nb8 Na5 85. Ka8 Nc6 86. Ra2 Nd8 87. Ka7 Rxb8 88. Ra1 Rc8

The captures:

It doesn’t matter which king reaches his position first however the captures still need to be taken in this order… because regardless of whether the white king reaches a7 first or second, the other white knight needs to take the black queen on g1 before it itself is taken otherwise there is no way for the bishop to reach g1… This also means the top half configuration has to be reached after the bottom half…

First of all the only thing that can and has to be captured by the pawn on b6 is a white knight. It has to be captured first in order to allow a black rook to escape and be captured by the c-pawn in order to allow the black king entry in the back rank to reach its final position and the rest of the captures…

The only piece that can be captured by the pawn on d3 is either the black rook or queen judging from the final position however the black queen plays a necessary role in the final position of the black bishop which entails that it needs to be captured after the king has infiltrated white’s back rank. It also needs to be captured before the remaining white knight because like the queen, the white knight is needed to aid the bishops entry after the king has infiltrated the back rank but it can only do so after an opening is created on c3…

Next the white knight has to capture the black queen on g1. The queen has to have been placed there beforehand in order to allow king f1 which is needed in order for the black bishop to reach its position via a queen capture by a knight and nh3 discovered check on the king… What led to this? This is because the bishop needs to come only after the king has reached this position since if it didn’t, it would entail four consecutive pieces from the right minimum in the back rank and so the king would be unable to reach the final position. This means the only way it can reach its position is by using this diagonal meaning the king has to get out of the way somehow, this entails the king going to f1 first but this needs two pieces blocking the white rook and queen. Firstly, neither the white rook nor the queen can escape to allow for this even before the king infiltrates so we need a piece on either side to block those pieces- the rook is blocked by the pawns and knight and the queen will not be able to re-enter again to reach h1 once the king infiltrates. The g1 square cannot have a white knight since it would check the king before hand and so it would either have to be taken or the king is mated. Nor could it have a black knight because the black knight would be pinned once the king reaches f1 and would have to be captured which then contradicts the final position. So the black queen has to find its way there somehow and once it does, the only thing that can capture it is the white knight because if the white queen does it instead the king is either mated or something has to capture the white queen.

Lastly

If the white knight isn’t placed on b8 before the black rook and knight reach their final positions, the king after reaching a1, would either be placed under check by the rook or be forced to go to a7 first to allow the black knight and rook to reach the configuration. However, this means the king would either be mated on f1 or the knight be captured since the knight can only reach d8 via c6… so the white knight is needed to be placed there to keep the king on a8 and allow the black knight to reach d8. After this, the white knight has to disappear. The black king and bishop would have to be immobile before the final steps leading to the completion of the top configuration, leaving only the rook and knight as the options to capture to white knight. The rook has to capture the knight on b8 and not the black knight since if it was the black knight doing so, any attempt to get the knight to d8 mates the king…

Other things that are part of the outline of any proof game:

In order to reach the bottom half configuration:

This configuration needs to be reached in order for the king to walk to f2:

Then

This configuration in the bottom is required to allow the Queen to enter the back rank:

All this is in order to allow this sequence of moves for the final black bishop and king positioning:

In order for the top half configuration:

The knight has to reach this position on h8 before pawn g6:

The knight has to block the rook like so on e8 to allow the king to infiltrate the back rank and it can’t be an f8 blockade either since that would trap the knight:

The king has to walk to a8 and then knight to b8 as a blockade:
From here it is just a matter of getting the rook and knight and king in the right place and then taking the white knight with the rook which has been already illustrated.

Answer 2

A retro-style solution (it seems to me easier there).

The sides have three captures each. Everything is clear with the c8 and f1 bishops. One capture is made by pawns (you can immediately forget about them). It is realistic to return only one white piece (knight) and one black piece (queen or rook) to the game.

And two kings must be freed. It's complicated down there. Let's start from the top. Here we need one white knight on b8. The black rook can give it to us. But not on c8. If the knight comes to b8 from the outside (from a6, for example), black will fall into the retropath.

Therefore, for example, so:

1. ... Rb8-c8 2. Ra2-a1 Rc8xNb8 3. Ka8-a7 Nc6-d8 4. Ka7-a8 Ne5-c6+

Further: bRc8-g8-g7; wNc6; wKa7-e7-f6

You can move on to the lower house. It will be more difficult here:

bNe5-c2-e1; bKf2-f1; wNc6-h3

1. ... Bc5-g1 2. Ng1-h3+ Rg8-g7

We got Black Bishop out. But now how to remove the knight on h3? With the king on f1, there will be an impossible check from the queen, with the king on f2, there will be an impossible check from the knight himself. The only way is to leave, leaving a black figure in his place. And this, of course, is not a rook. So, the queen.

3. Nh3xQg1 Kf2-f1 4. Nf4-h3 Nc2-e1 5. Ke5-f6 Nd4-c2 6. Nd5-f4 e7-e6 7. Ne3-d5 Ne6-d4 8. Nc2-e3 Ng5-e6 9. Ke4-e5 Nh3-g5+ 10. ~~ Qe1-g1 11. ~~ Ng1-h3 12. ~~ Kf1-f2 13. ~~ Qg3-e1 14. Ne1-c2+ Kf2-f1 15. Nc2-e1 Nh3-g1 16. Rg1-d1 ~~ 17. Ne3-c2 Ke1-f2 18. Nf1-e3+

The Black King is going home. Then you can return c2xRd3. The black rook goes to a8. You can return a7xNb6.