Anything besides 1...gxf3 and black will win. 

But whatif black doesn't take the pawn ??

 blame the book

Blagy -- that  4...g3?? is a nice cautionary tale! How easy to throw away a win with careless play... and if instead Black had bothered to look for better moves he would have been able to end the game very quickly.

U Should look for  more ways that the opponent can save itself

Ryalmacin, the author of the "puzzle" is only rated 1049.

One of the definition of "Hogwash" in the English dictionary is "Garbage fed to the Hogs".

After 1.f4 Kxf4 2.Rg1 the simplest win for Black is with 2...Qh2 See diagram

A. 3.Ke1 Kf3 4.Rf1+ Ke3 winning

B. 3.Re1 Qb1+ 4.Ke2 Qxe1+ 5.Kxe1 Kg3 followed by 6....Kh2 and the queening of the g-pawn wins.

C. 3.Rf1+ Ke3 4.Re1+ Kd3 wins

In a K+Q vs. K ending, the Q can, without the help of the K, force the enemy K to the edge of the board where the enemy K is checkmated.

The trick is to constrict the moving space of the enemy K each time the K moves.

Here's an example:

Mate in 5:

HAHAH

Nice one!