Ronald Turnbull's Solution to Dead Reckoning original E:

All missing black units have been captured by white pawns, and White's g and h pawns have died on file without capturing. Apart from those two pawns, all missing white units have been captured by black pawns. Black's last move was not a capture. It wasn't Pc6-c5 as this pawn came from a7. Light-square wB died by Pe6xf5, so the pawn now on f4 came from e7, and the pawn now on f4 came from c7. So Black didn't just move this e-pawn, nor did he move K from e7 (improper check).

The diagram itself is a dead position. Black's last move wasn't with f-pawn or Rook as these moves were from dead positions. The same goes for any move to the diagram by black King - except for Ke8-f8. If Black just made that move, and if he can still castle, the position before that move was alive: he could have played 0-0 followed by wPc3-c4; bRd8; Pc7xd8 with a live game after the promotion. As this is the only way there could have been a legal move to the diagram, we know that Black did play Ke8-f8 from a position where he could have castled.

The Black King has moved just once.