So, two White pawns are gone: the a-pawn and either the d- or e-pawn. Also, a Black pawn (either d- or e-pawn) is gone along with a knight and a rook. The White rook on h8 may have captured the other Black rook there... but I think the biggest question is how it got there. I think White may have had the opportunity to castle at some point to get his rook into position to move to the eighth rank... however, if that's the case, there go AT LEAST four moves immediately (0-0, Re1 or Rd1, Kf1, Ke1).
Also, I can think of a few more moves... after White's dark-squared bishop moved out of c1, we can have Nc3 followed by Rc1. The knight on b1 may well have gotten there via Ng1-f3-d2-b1.
Of course, now that I think about it, White's rook on h8 may in fact be the one originating at a1. Castling kingside is still a possibility though, and would involve 0-0, Rc1, Kf1 and Ke1. Meanwhile, the a1 rook could have a track of Ra1-a5-d5-d8-h8, but that's quite a list for 17.5 moves.
Here's a list of possible moves I've come up with for White as of right now:
0-0
Rc1
Kf1
Ke1
Nf3, Nd2, Nb1
Nc3
Bg5, Bh4
Bc4
exd3
Qg4, Qb4
Ra5, Rd5, Rd8, Rh8
This puts us at 18 White moves, which fits into the stipulations of the puzzle.
Here's a third exact proof game...
An "exact proof game" is a type of retrograde chess problem in which the solver attempts to completely reconstruct, move by move, the history of the game.
Here's the position. From this position, and the stipulation that it has been reached in a regular game of chess in 17.5 moves, the challenge is to completely reconstruct the game's move history. ("17.5 moves" means that the position shown is immediately after white's 18th move.) Good luck.
Please feel free to add comments as you make progress toward a solution, so that the collection of insights into this problem grows.
Note: I did not compose this problem. I will reveal the author when it is solved.
Amy