Can white win in this almost 20 year old puzzle? It's White to move-can they force a win?. The left bottom is a1 as usual.
[FEN &quot;4k2r/6p1/p4pKb/p2N3p/p7/p5Q1/p1n5/qrn5 w - - 9 9&quot;]

Note for people who saw an earlier version: the position was edited (including a swap of the colors) for a more beautiful puzzle by Laska's suggestions. The answer has now been fixed accordingly.
History: I came up with this puzzle when I was a student, almost 20 years ago. It was published in the magazine of my local chess club, but nobody found the answer. A friend popularized it with his new chess-friends after he emigrated to Canada.
The solution is


   1. Qc7 followed by Qe7, Qf7 or Qc8 mate (depending on Black's second move).


This works because


   Black can't castle because either his king or rook on h8 have moved already. As Brian Towers notes, the black a-pawns need to have captured 10 white pieces and/or pawns; there are enough pieces missing, but capturing the white e-pawn (or to the right) doesn't bring you any closer to the a-file. So, one white pawn on the queenside must have been captured on another file; this is possible via promotion, but since there's just one black piece missing (a white-squared bishop) the promotion could have happened on the d-, e-, f-, or h-file. The pawn has either attacked the black king at f7 or d7, or it promoted on e8, forcing the black King to move in all cases. Or it promoted on h8, forcing the black rook to move. Another possibility, as @Laska noted, is that the e-pawn captures the black bishop on the d-file and is captured itself before promotion; however, we only need a single capture on the d-file, and the original white d-pawn must be captured too.