First glance it didn't seem to tricky. There can't be any promoted men. Only the f7 pawn has been taken, but that must have been captured there.
White needs 7 captures (2N, 2R, Q, wB and fP)
Black can only make two (R and Q). These must be d7xe6 and axb(not yet sure what.
Looks as though the f1 and f8 bishops must be firstly captured by Ns too, adding to a basic 12 moves for Ns just to reach their final positions if a direct route can be taken.
Problem is that to unblock e6 the B on c8 must be replaced. Only two uncaptures possible for white prior to this, on c6 and d6. Therefore the latter must be the B. It would be easy if two rooks can be uncaptured as I'd want to get both through c8 before the B returns.
Right, I've retracted back to this position. I'll give the moves a little later, but as white's uncaptures will give black two pieces (R & Q) which are free to waste a move, alongside the wR, the Ns can go about their business.
Both f bishops must be captured upon their starting squares by knights, though in no case do I have to worry about putting the K in check upon approach. And both capturing Ns can move onto their finishing squares without interrupting other crucial play.
Only problem with this is that whites last uncapture (a2xb3) will close his position so must move Ns in order to allow black to retreat the final few moves as quickly as possible.
And conveniently I can make the basic N moves which don't block the position whilst black takes back the final three N moves.
From here white uncaptures to leave a black N on b3, three moves back for each side.
This can be reached as the black c or a pawn can be left on the fourth rank after my retractions (to follow) in the first instance. Plently of opportunity to waste time still available.
Therefore total moves:
bN b8-c6-d4-b3 (3 moves)
bN g8-f6-g4-e3xf1-e3-d5 (6 moves)
wN b1-c3-a4 (2 moves)
wN g1-h3-f4-g6xf8-e6-c7-a8 (7 moves)
Total of 18 moves then.
Edit: I can't go via e6 and c7 as I'd check the K on the way. e6-f5-e6 would solve the problem with two extra moves (to make 20)
It is possible for the N to move from f8 to a4 and the other from b2 to a8 but these both take the same number of moves.
My answer is twenty, not 18.
Originally posted by PeakiteCorrect. Good spot on the Nc7+ issue.
And conveniently I can make the basic N moves which don't block the position whilst black takes back the final three N moves.
[fen]r1bqkbnr/pppppppp/8/8/N7/1P5N/1PPPPPPP/R1BQKB1R w KQkq[/fen]
From here white uncaptures to leave a black N on b3, three moves back for each side.
This can be reached as the black c or a pawn can be left on the fourth rank ...[text shortened]... r from b2 to a8 but these both take the same number of moves.
My answer is twenty, not 18.