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Very tricky to work out kings route to f8. I guess that is the point of this puzzle.
White's c-pawn clearly captured by dxc6 and White's queen by Rxg8#
Last few moves were 17. ... Nh6+ 18. Qg8 Rxg8#
Before then... perhaps 16. ... Be8+ 17. Kf8
Minimum number of white moves needed seems to be exactly 18:
c2-c4-c5-c6 (three moves)
Na6
Bb2-c3-b4 (three moves)
Rc1
Qb3-g8 (two moves)
Kings trip to f8 (eight moves)
So Blacks e and f pawns must have been captured by the white king.
Problem - if White's last move was Qb3-g8 - how did his pawn end up on b3? I must be wrong - Qg8 was played earlier and the only move we know for certain is 18. ... Rxg8#