Once again, here is a new retrograde chess puzzle! Recently, I have been trying to build something interesting with all 32 pieces still on the board. In this attempt at that, White somehow comes out victorious by delivering a checkmate at the end. Exactly how did that happen?
What was the final move?
(16+16) FEN: 5bnn/4prRb/5prq/1p1N2pk/RPNp2Qp/p1p3PP/P1PPPPBB/6K1
Please provide your reasoning in your answer. Have fun solving this! :^)
Looks like the last move must have been
Queen from e4 to g4, checkmate.
That's because the top right clump is completely frozen in place, and the last piece to move in must have been the black king.
At the time that happened,
and that's the clincher. After the top right gets sealed in, white still needs, at the very minimum,
and all of this has to happen before the final, mating move.
Contrasting this with black only having 4 pawns to move, one of which must already have advanced to let black's light square bishop out, the timing is going to get really tight, and indeed the white queen will have only one move to reach the mating square.
At first, seems like there might be two ways to achieve this: with the white queen at c8 (blocked by black's d-pawn while the king is at g4), or with the white queen at e4 (blocked by white's bishop at f4). At any other place the queen would be either blocking the black pawns, or threatening g4, preventing the black king from being there with white to move.
However, trying to construct the critical position for the first option, it turns out the knights are Very Hard to place:
Wherever the knight that ends up at d5 is, it will either block the rook's path, or hit one of the black king's squares. And the stupid (by which I mean painfully clever) thing is that a knight standing at c3 isn't what blocks the rook's path, it's that having to move the knight to d5 before the rook has gone by is a problem: looking at the goal position, black's final move can only have been "pawn to b5", so black's b-pawn must stubbornly stand at b6 while white is manoeuvering the rook, and thus
the fifth rank was the only possible route for the white rook to reach the a-file (behind the black a-pawn) in time.
So the latter option seems to be the only way to go, because now we can make one more square available for the troublesome knight:
1. b4 d6! 2. Na3 h5 3. Rb1 h4 4. Rb3 Bf5 5. Rg3 Qd7 6. Rg6 Rh5 7. Rh6 Nc6 8. Nf3 Ne5 9. Rh7 Ng6 10. Ne5 Nh8 11. Nd3 g5 12. Nb2 Qc6 13. Na4 Kd7 14. Bb2 Bg7 15. Be5 f6 16. Bf4 Rf8 17. Qa1 Rf7 18. Qb1 Bf8 19. Rg7 Bh7 20. Qa1 Rh6 21. Qb2 Rg6 22. Qb3 Qf3 23. Qc3 Qg4 24. Qb2 Qh5 25. Qc3 Qh6 26. Nb6+ Ke6 27. Qd4 Kf5 28. Qe4+ Kg4 29. h3+
and indeed from here we can reach the final position:
29. -- Kh5 30. Bh2 c6 31. g3 c5 32. Bg2 c4 33. O-O d5 34. Rb1 d4 35. Rb3 a6 36. Rf3 c3 37. Rf5 a5 38. Nac4 a4 39. Ra5 a3 40. Nd5 b6 41. Ra4 b5 42. Qg4#
Here's the full proof game on Lichess.
INCORRECT ANSWER. I noted my mistake below.
Obviously the final move was the white queen moving from somewhere to g4. But where did it move from?
The white pieces in the lower right can only come about with white moves h2-h3, B??-h2, g2-g3, Bf1-g2, in order. (Pieces could move around and come back after that.)
The white h1 rook used at least 6 moves after the above Bf1-g2 to get where it is now. This rook can't get out past the white king until after Bf1-g2, plus at least one more white move (such as 0-0).
So white has made at least these 11 moves after h2-h3: B??-h2, g3, Bf1-g2, 0-0 (or more), 6 rook moves, Q??-g4++
Since no pieces have been captured, we can unambiguously talk about the black pawn on a given file. The last black move which was not a pawn push was Kg4-h5. This happened in response to white h2-h3 or earlier, so black has made at least 10 pawn moves since then. At least one of those moves pushed black's c pawn, and the white knight moved to c4 after that. Now we know white made at least 12 moves after h2-h3, so black made at least 11 pawn moves. At least one of those moves pushed black's d pawn, and the white knight moved to d5 after that.
So white has made at least these 13 moves after h2-h3: B??-h2, g3, Bf1-g2, 0-0 (or more), 6 rook moves and 2 knight moves, Q??-g4++.
Before black Kg4-h5, black moved either the b7 or d7 pawn (or both), to let out the c8 bishop. After Kg4-h5, black made at most 12 moves, all of them pushing the a, b, c, or d pawn. Given the earlier b or d move, only 12 pawn moves are possible.
Therefore the game ended with white h2-h3+, black Kg4-h5, exactly 12 white moves and 12 black pawn pushes, and finally white Q??-g4++.
Before h2-h3+ Kg4-h5, the white queen was not threatening g4. The queen made only one move after that, onto g4. Therefore a piece was between the white queen and g4. The white rooks were stuck on the first rank and g7, so it couldn't be either of them. The white bishop at g2 and the knight at c4 each made just one move after that, and couldn't have been in a place to block the white queen.
There was my mistake. The bishop could in fact have been at f4, and that's the case in the only actual solution.
If the white knight at d5 moved from f4, that could block the queen at e4, but that's not possible since the knight at f4 would make Kg4-h5 illegal.
The only remaining possibility is that the white queen was at c8, blocked from threatening g4 by the black pawn at d7. So the final move was Qc8-g4++.
