Since the puzzle turned out to be way hairier than expected, my earlier answer got quite messy and hard to follow, so here's a complete rewrite. It's still very clunky, but I hope it's much clearer than the earlier one. Also, there's an executive summary at the end, if you are not interested in all the gory details.
First, we'll need to make some general observations. If they seem to be overly complex, try to hang on: every single one of these will turn out to be important. (Yes, really!)
Observation 1: White has 15 pieces on board, and black has a doubled pawn on the b-file, and a pawn on the c-file.
Consequence 1: The only white piece that was captured was taken by black's a-pawn. No other white pieces were taken.
Consequence 1.1: The five missing black pawns were either taken or promoted on the file they started on.
Observation 2: Assuming minimal captures, white's pawns are all identifiable, up to the pawns on d-file being interchangeable. The only missing white pawn is the h-pawn.
Consequence 2: The piece taken by black's a-pawn was white's promoted h-pawn. (Or another white piece, and the h-pawn was promoted to replace it. Since it doesn't matter in which order the capturing and the promoting happened, I'll just keep talking about "capturing promoted pawns" when I mean both of these possibilities.)
Observation 3: White's b-pawn has ended up on the d-file, and black's c-pawn is still on the board. Also, all black's non-pawn pieces are still on the board.
Consequence 3: White's b-pawn has taken a promoted black pawn on the c-file, and the black d-pawn on the d-file. (Either that, or two promoted black pawns. The former case is the relevant one though.)
Observation 4: White's g-pawn has taken black's f-pawn on f3.
Summary so far: Out of the 5 missing black pawns, we have accounted for 2 captured as pawns (on d and f files) and one captured promoted on the c-file.
Observation 5: In order to promote the h-pawn, white must have cleared the path somehow. There are three options:
1. white took black's h-pawn
2. white's h-pawn took black's g-pawn
3. white's h-pawn took another black piece on the g-file, and a pawn was promoted to replace it.
Consequence 5: no matter which of the three cases occurred, black lost one promoted/promotable pawn.
Observation 6: black's e-pawn was always blocked from promoting. The above cases have accounted for all the other black pieces, so it was impossible for white's d- and e-pawns to have swapped places by capturing, which could have opened the way for the e-pawn to promote.
Consequence 6: Either one or two black pawns were promoted. If two were promoted, one of them was captured on the g-file. In either case, one promoted pawn was captured by white's b-pawn on the c-file. In other words, all promoted black pawns have been captured by white pawns.
And now, FINALLY, we are ready to tackle the puzzle!
At first glance, it looks like
could be the solution. Sure, why not. Let's check that the position could have been reached by legal play, though!
The black pieces are pretty much clumped up, so we can list all black's possible last moves:
- Queen moved to c1 - impossible, because it would have started on a square where it was giving check
- Pawn d2 promoted to Queen on c1 - impossible, because it would require a capture (see "consequence 1")
- Knight took on b4 - impossible, requires capture.
- Knight moved to empty b4 - seems plausible. But let's investigate further!
If black's last move was Nb4, then it follows that white's previous move must have been Bxc3+. If not, then the white bishop would have been giving check on white's turn, which is of course illegal. But this poses a problem: there cannot have been anything on c3 for the white bishop to capture, because above we have accounted for all promoted black pieces (see "consequence 6"), and the only pawn that could have reached that square is still on c4.
From all this, is follows that
It cannot be white's turn!
Well, where is the mate then?
Given the information from above, we can try to figure out white's last move. It must be something that enables a new previous move for black, since if it doesn't, we know from above that black has no moves.
Let's explore the four possibilities for this case:
Possibility 1: White's last move was Nb6. This enables a pawn move for black's previous move, but then the white move before that is impossible; there's no way the white rook can be giving check:
- the rook cannot have moved to c5 to give check
- the white pawn on a6 cannot have been on b5, discovering the check, because it would require a capture, and all black pieces are accounted for. (Black's only "unaccounted for" piece is the e-pawn, which cannot have left the e-file.)
- and obviously, white's queen couldn't have revealed the check either.
Possibility 2: White moved something that enabled the black queen to move last
- sadly, no such moves exist.
Possibility 3: White took black's e-pawn, which was black's previous piece to move
- Since black's e-pawn cannot possibly have left the e-file (see conclusion 1),
the only square where this capture could have happened is e3, and the capturer could only have been white's d-pawn. That would mean that one of the white pawns on the d-file is actually white's e-pawn. That could only happen if one more black piece was captured, which we already know didn't happen, so this line is also impossible.
Possibility 4: White moved something that enabled the black knight to move last
This looks promising, but there's still the problem of finding a suitable move.
- It could not have been a bishop move, because the bishop would still have been on the black king's diagonal
- It could not have been pawn d3 to d4, because now the knight has no square from which to jump into its current position
- So it must be pawn d2 to d4.
This enables a move for the black knight, because it opens up a new previous move for white: now the bishop can have come from somewhere that's not on the black king's diagonal.
In other words, three half-moves ago, the board must have looked like this:
(white's dark square bishop could also be elsewhere on the long diagonal)
From there, the only way to reach the current position is
1. Bc3+ - Nb4
Which finally allows for the long sought after mate in one:
2. - cxd3 e.p.#!
Phew! What a ride!
At the first glance, it looks like
1. Nxc4#might be the answer. However, a very complicated analysis proves that it cannot be white's turn to play. Starting again with the assumption that it's blacks turn, another round of analysis proves that white's last move can only have been "pawn from d2 to d4". This enables black's c-pawn to capture en passant, which checkmates the white king.
I may have missed some specific variations, so please drop a comment if you notice one. And thanks again, OP, for another brilliant puzzle!