Polgar 4013

A mate in three from Polgar "Chess", #4013

If g3 were not pinned, g4 would be mate in one, so the queen is tied to the third rank unless it can give check. Therefore the rook is free to threaten mate in one also (on h7). The queen is overworked -- cannot both maintain the mate-preventing pin on g3 and also remove a mate-threatening rook.

So why isn't it mate in two? Because the queen can sacrifice herself to gain one extra move (Qxg3+).

And can't the rook threaten mate from either h8 or g7? Yes, but one of those moves (Rg7) also allows black to delay mate by a move (with h6).

1 Rh8 Qxg3+ 2 Kxg3 h6 3 Rxh6#