Originally posted by XanthosNZ
Well apart from the fact that it can't be a black queen or rook I have no clue where to start on this.
White has made the last move: pawn c7xNd8R+. There had to be a white pawn on c7 otherwise black's last move left him in check.
It could not have been a queen or rook on d8, because the white king would have been in check since there is no way that black's last move was to play away a piece from the back rank standing between the queen and white king (it would still be there one move distance from that square). It was not a bishop on d8 either: white square bishop obviously not, wrong colour; black squared bishop has never moved from f8 because the pawns are still there on e7 and g7. So it was a knight.
That means that black promoted one pawn to a knight (the other two still on the board). With the h-pawn. The f-pawn is on c4.
Therefor, the piece on h4 is not a pawn, all 8 are identified elswhere. It is not a black knight either for the same reason. Not a black bishop (the black squared was captured on f8, and the white squared is wrong colour). It cannot be a black rook or queen because the white king was not in check. That eliminates all black pieces.
So it is a white piece. Black captured five pieces, leaving still the last one on h4: 1 axb6; 3 with the f-pawn to c4; and 1 with the h-pawn. Only possibility is from h3 to g2 to promote on g1 (all other possibilities require at least two captures). That means all captures were on white squares. Hence, the remaining piece is the white black squared bishop on h4 (thank god that is a black square).