N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
1st of all, how do you know that white's K was at e1 last move. Furthermore, why did the knight have to capture something on f1 or e3?
white's last move was either 0-0-0 or by the missing piece at g4 - K move from check of immobile B not possible. Now if there is a missing white piece at g4, then neither of the N, B(h2), Q has captured anything - which means black B went to h2 first, then white played g3, then the white B somehow managed to enter to h1, and the black Q followed the white bishop all the time exposed to its attack to g2 without capturing anything - does not make sense, does it? So there is no missing white piece at g4, so white's last move must have been 0-0-0, so white's king must have been at e1 last move. If that was so, how can black's supposed e-pawn promotion have taken place - exf1=Q#
If the N have not captured anything, then is there any valid logic it went to e3 in front of 2 pawns (it had not given check there), then to f1?
You seem to be missing the point; it isn't about finding what moves were played because they were good moves... it's about finding what the possible moves were that led to the position... so hanging pieces is perfectly reasonable. If your only reasoning for theqeen not being allowed to sit on g2 for so long is that white could capture it every move, you have no valid reason in a deductive puzzle whatsoever.
N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
1st of all, how do you know that white's K was at e1 last move. Furthermore, why did the knight have to capture something on f1 or e3?
white's last move was either 0-0-0 or by the missing piece at g4 - K move from check of immobile B not possible. Now if there is a missing white piece at g4, then neither of the N, B(h2), Q has captured anything - which means black B went to h2 first, then white played g3, then the white B somehow managed to enter to h1, and the black Q followed the white bishop all the time exposed to its attack to g2 without capturing anything - does not make sense, does it? So there is no missing white piece at g4, so white's last move must have been 0-0-0, so white's king must have been at e1 last move. If that was so, how can black's supposed e-pawn promotion have taken place - exf1=Q#
If the N have not captured anything, then is there any valid logic it went to e3 in front of 2 pawns (it had not given check there), then to f1?
I'm convinced that the solution to this problem will not invoke likelihoods, but rather certainties. If it's not proveable it's not the solution.