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Yes, but there is a better way to win rather then to draw.
Let's apply some logic here...
To successfully solve this problem to the limitation presented, in exactly one move, you would have to give check and cover the following squares: g8, h8, g7, h7, g6, h6.
The king currently covers the g6 and h6 squares, and since it cannot give check, it can't move.
Moving the pawn uncovers the rook to attack h7. But no matter what you promote the pawn to, the g8 square is still unprotected: the King would be able to recapture that new piece. Moving the pawn isn't an option.
No matter where you move the Rook, it cannot give check, so it can't be the Rook.
The Knight can give check from f8, but the g8 square would still be unprotected. Thus moving the Knight isn't an option.
Therefore, this problem cannot be solved according to the limitations presented, there is no mate in one.
I see what you did there!
Alright. Let me guess. The board is flipped 90 degrees clockwise and a piece has already been lifted from a8 (h8 in diagram) so that the pawn can go there and mate as a queen.
So, the whole premise of this "puzzle" is to ignore the rule of promoting a pawn to a piece of the same color?
The mind truly boggles.
* Goes for beer *
(It's early but I've been driven to it)
p.s. That's better, all makes sense now!
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