The solution:
The board is rotated 90 degrees clockwise - upper right-corner is a8, bottom right-hand corner is h8.
So mate is achieved by cxb6 e.p. #
[clue: bottom right-hand corner should be light square, not dark].
Why did we assume black's last move was b7-b5?
Black's last move could only be b7-b5. R, 2 Bs, couldn't have moved. N couldn't have moved for then white could have been in check while black moved.
Pawn moves: can the g2, f4, f5 pawns have moved?
2 white pieces and 4 white pawns are missing. So g2 pawn was originally h7-pawn (if c7 pawn, more than 6 captures total involved), cannot have moved. f5 pawn must have been d7 pawn, cannot have moved. f4 pawn must have been c7-pawn, cannot have moved. Thus b-pawn only has moved. It cannot have moved from b6, as white king would have been in check, so must have moved from b7. QED
This one isn't from the Deductive puzzle's I have been doing, but rather something I just stumbled upon elsewhere, and just had to share with you all.
"White to mate on the move."
http://www.futilitycloset.com/2009/07/25/christopher-crustys-little-idea/