oh. that's amazing :) thanks for the hint, blauesross :)

i am so happy to see other guys than me have also problems solving this puzzle ;-)

when i first found this problem i thought about it for over half an hour and like u i came to the (wrong!) conclusion that it is impossible to reach the position after exactly 4 moves, however i was wrong

u made a lot of interesting arguing, however ur first assumption is wrong:

"so at most one pawn has changed file, leaving at most one vacant file. "

Maybe u want to try again, i assure u it is not a trick and u have overlooked one possibility in ur reflections ...

in ur last paragraph lies an error and the key to the solution:

"The pawn capture cannot occur earlier than white's 3rd play (opens with pawn, moves bishop, captures pawn), so the pair of moves to capture and return must be black's 3rd and 4th.   Whichever piece was used, this forces the two pawn moves c6 and e6 to be first and second, in either order, before the move B x d7 occurs on move 3.   But if c6 and e6 have already been played, B x d7 cannot be played as the diagonals to d7 are blocked."

Does this work? I would have to suggest that 'no, I cannot find out how, because it is not possible in the legal moves to chess to arrive at this position in 8 semi-moves.'   Either I'm wrong or if I'm right it's a trick question so I don't mind posting the 'truth' to save people from going crazy!

There has been exactly one capture of a white chessman, the white-square bishop, so at most one pawn has changed file, leaving at most one vacant file.  So if the bishop was taken by a pawn it must have been the d-pawn.

This would have to have occurred on c6, c5, c4, e6, e5 or e4 (all the squares accessible from a d-pawn on d7, d6 or d5).  Since white must play the e-pawn at least once, and the light-squared bishop out at least once, he has at most two moves spare for any piece other than the e-pawn and light-squared bishop, and no piece can reach c6, c5, c4, e6, e5 or e4 in a move and return to its starting position in the white camp, so no white recapture of the bishop would be possible.   If the d-pawn captured it, it must still be on the board.  

The d-pawn would still be present on c5 or e5 if it captured there, which it is not.  Therefore if the d-pawn captured the bishop, the only possible pawn to have done so, it must in fact now be on c6 or e6.   And since no pawns are on c7 or e7, then whichever file housed the d-pawn now, is missing its original pawn, so the bishop must itself have captured on c6 or e6 respectively.   But either way we have at least 3 pawn moves - c, d and e pawns - but also at most 3 pawn moves, since the d-recapture is a single move and the c and e advances are by single moves.   So exactly 3 pawn moves would be required to have occurred assuming a pawn captured the bishop.

But then there is a single spare move which cannot be used to pass, and cannot be used to both move and return a piece to its original square.  Therefore the pawn capture hypothesis is unacceptable.  The bishop must have been captured by a piece and not by a pawn.  So no pawns have changed files, meaning pawn plays of the c-pawn to c6 and the e-pawn to e6 and either 0, 1 or 2 plays with the d-pawn in its own file. Since the d-pawn is no longer on the board, it has been captured and only the bishop has had time for the capture, so the bishop must have taken it on d7 or d5 (light-squares).   But d5 is not possible because it will take 2 moves for a piece - the queen in this case - to capture it and return to the home square, but we have already required 3 pawn moves in total (c6, d5 and e6).   So the d-pawn must have been captured on d7, allowing c6, e6, and  then the bishop capture in one of three ways, [Q, QB or K x d7], followed by Q, QB or K back to their original square. 

The pawn capture cannot occur earlier than white's 3rd play (opens with pawn, moves bishop, captures pawn), so the pair of moves to capture and return must be black's 3rd and 4th.   Whichever piece was used, this forces the two pawn moves c6 and e6 to be first and second, in either order, before the move B x d7 occurs on move 3.   But if c6 and e6 have already been played, B x d7 cannot be played as the diagonals to d7 are blocked.   

This does not leave any possible histories for this position in exactly 4 moves.