Can you find the last 20 moves? #2

Question

In this puzzle I finally managed to implement some themes that I really like, but so far was not able to get working properly in any of my own puzzles. Composing this was quite the challenge and in the end I'm very pleased with how it turned out:

Can you determine the last 20 single moves that lead to this position?

As usual, please provide your reasoning and a proof game in your answer. Have fun! :)

Answer 1

The solution is at the last of the following imaginary game. Some few comments on how the solution was found follow. After seeing the solution, simply contemplating how the moves work one after the other one make any comment superfluous. The discovery-check idea was realized here in a superb manner, the retrograde analysis moreover finds a unique path of this piece from its uncapture to its critical field h6. The hidden theme is nicely implemented, the unique path of the two white bishops to their initial squares!

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First of all some comments that lead to the solution.

• All white and black pawns and on the board. Else white misses two pieces, the Bf1 and a knight, and black lost through captures the two knights, and the queen.
• In the given situation white and black have some few obvious moves to retract, but then no moves can be taken back any longer. So we need a quick scheme to unravel the situation of the kings, and/or unblock the accumulation of pieces around the h1 corner. The key is to see how the Bh6 could come in there, this is the key. The only possibility is a discovered check from Rh7 to Kh5. Which piece can discover the rook line? Only a knight. So we desperately need in the minimal number of moves a black knight to land on h6. The quick path without wasting any other moves (as seen later, we have no time to waste pawn moves) is unique, thanks to the talent of he composer, it is Nb5-d6-f5-h6. And it is Nb5, not Nb3. (The Nb3 needs the d4 square, that pawn has to go back, but this little move is missing when the puzzle of retracted moves is finally checked.)
• White misses also the Bf1. The pawn f3 must come from e2 and needs one capture for this. Before capturing, the white light square bishop must be on f1. Then g4 must have been the h-pawn, one more capture needed. There are two pawns on the b-file. So there is at least one capture with one of them. All white pawns and black pawns are on the board. Since exactly 3 black pieces are missing we roughly know which were the captures: (1) e2xf3, (2) h2xg3 or h3xg4, and (3) c2xb3 or c3xb4 or c4xb5. But as mentioned, the b5 is the critical square, where the black night comes back in the retrospective.

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Solution:

lichess link for the solution

[Event "PUZZLE Tim Seifert https://puzzling.stackexchange.com/questions/127291/can-you-find-the-last-20-move: Kapitel 4"]
[Site "https://lichess.org/study/JwEcp3In/hC7jHGui"]
[Result "*"]
[Variant "From Position"]
[Annotator "https://lichess.org/@/dan_fulea"]
[UTCDate "2024.07.04"]
[UTCTime "16:45:51"]

1. a4 b6 2. Nf3 Nf6 3. Ng1 Nh5 4. Nf3 { White needs to lose some moves, so this night moves are just chasing time. The solution was elaborated backwards, so please do not search for a particualar reason. } 4... Ng3 5. Ng1 h5 6. Nh3 { This knight goes now as quick as possible to f6, so that after gf6 black can shuffle the pieces from the last rank. } 6... Nc6 7. Nf4 Nd4 8. Nd5 Rh7 9. Nf6+ gxf6 { The white knight has its mission accomplished, the black pieces will get out of the last rank now, trying to reach as quick as possible their final positions. } 10. Nc3 Bh6 11. Nb1 { White further needs some waiting moves, till the black pieces reach their better places. } 11... Kf8 12. Nc3 Kg8 13. Nb1 Kh8 14. Nc3 Qg8 15. Ne4 Qg5 16. c3 Qa5 17. Ra3 Ndf5 18. Rb3 Bb7 19. Rb4 Rg8 20. Rd4 Rg5 21. Qb3 Bg7 22. hxg3 Nh6 23. Kd1 Ng8 24. Rh4 Bh6 25. Rg4 Rf5 26. Rg7 Rg5 27. Kc2 h4 28. Kd3 Qa6+ 29. Ke3 Rg6+ 30. Kf3 Bg5 31. Kg4 Nh6+ 32. Kh5 Bd5 33. Rd3 Be6 34. Qb4 Bh3 35. g4 Qc4 36. Ng3 Bf4 37. Nh1 Bh2 38. Qd6 Bg1 39. Qh2 Qc6 40. Rg3 Qf3 { And now there is a unique path from here to the final position, seen in both directions. } 41. exf3 b5 42. Bd3 b4 43. Bb1 b3 44. Ba2 bxa2 45. b3 d6 46. Ba3 d5 47. Bd6 d4 48. Bf4 Nf5+ 49. Bh6 Nd6 50. c4 Nb5 51. cxb5 *

Answer 2

Partial answer

Let's start with:

Pawns

There are:

All of them, 8 for each side. No pawn was lost nor promoted.

Hence:

1. The black pawn from G7 captured something in order to land in F6. So gxf6 was a black move.

2. The white pawn from E2 captured something in order to land on F3. So exf3 was a white move.

3. The white pawn from H2 captured something in order to land on G4. I'll detail that later below.

4. The black pawn from B7 captured something in order to land on A2. It also walked a lot and outmanuevered the white pawns somehow, which is only possible if the capture happened before the B white pawns advanced but after the A white pawn advanced.

   So, the possibilities are:

   The first two or three black moves for that pawn are b6 (maybe), b5 and b4.

   And:

   The last two were b3 and bxa2 or bxa3 and a2.

   That is:

   Four or five black moves.

   Also:

   White did a4 or perhaps a3 and a4 before that black pawn captured something. One or two white moves here.

5. The white pawn from C2 captured something. It is either on B3 or B5 now, the other being the white pawn from B2. One of them also walked a few squares.

   So, the possibilities are:

   A: cxb3, b4, b5, plus a b3 somewhere after the b4. Total of 4 moves.

   B: c3, cxb4, b5, plus a b3 somewhere. Total of 4 moves.

   C: c3, c4, cxb5, plus a b3 somewhere. Total of 4 moves.

   D: c4, cxb5, plus a b3 somewhere. Total of 3 moves.

6. The black pawn on D4 is easy: It is either d3, d4 and d5 or d4 and d5. No captures. Two or three moves.

Anyway:

There were 5 pawns capturing something (3 white capturing black and 2 black capturing white).

And what is missing:

7. Both black knights, the black queen, the white bishop on light squares, one white knight - 5 pieces.

   Hence:

   Each one of those pieces was captured by a pawn. No pawn captured twice. No other captures happened.

   Further:

   Each one of those must necessarily have been moved at least once in order to be captured. Otherwise, no pawn could reach them.

   Also:

   The black pawn on F6 couldn't capture the white bishop due to square's color mismatch. So it captured a knight, that needed at least three moves to get there. The other black pawn must have captured the bishop and this means that its last two moves were bxa3 and a2, also that bishop must have moved at least twice.

   Let's see more moves:

   The black queen must had at least two moves before dieing. Further, at least two moves for the king-side black knight and at least three for the queen-side black knight.

There are some strangely placed pieces:

8. The black bishop on G1.

   Why?

   It could only come from H2 before it was blocked by a lot of white pieces. Hence Bg1 was a black move. Sometime before that, it also needed Bh2 and before that either Bh6 and Bf4 or Bg7 and Be5. At least 4 moves (possible more because it could move around somewhere else before going to where it went).

9. The white knight.

   Why?

   It could only come from G3. Hence Ng3 was a white move and Nh1 after that. And there must be at least one other move before (Ne2 if it was exactly one).

10. The black bishop on H3.

    Why?

    It could only reach that before the white pawn on G4 reached that place. So, Bh3 was a black move before g4 was a white move and hxg3 was another white move before.

11. The white king.

    So:

    How the heck it got there? It would need at least 4 king moves to get there. But in fact, it was actually at least 5 since it needed to go around the pawn at F3.

12. The white rooks.

    Which is:

    At least 4 moves.

13. Other black pieces.

    Which is:

    3 king moves and 3 rook moves if there was no castling or castling and a bunch of other moves.

14. The white bishop.

    So:

    It is in the same diagonal from its home square, but there is an unmoved pawn blocking the way. So it must have moved several times around it to reach this place. At least 4 moves.

So far we have:

29 confirmed white moves and 25 confirmed black moves.

A curious note:

Those two rooks on G6 and G7 are on the opposite sides than what would be expected. Putting them there without mating the kings is complicated.

This is not enough to fully reconstruct what is going on, but I think that it is at least a good partial answer.

Hopefully this graph will be helpful:

That is...

A partial topological sort of which move must happen before some other move. By adding all the moves there, we then would need to just find which order from the starting nodes going down to the moves on the end nodes that works.