@Mcabion and vengene69... I think the point of these puzzles isn't to find the easy way, such as continual check into a mate, rather to help folks understand thier chessboard better. Better and Best chess is played by people who understand not only things like checks and captures, but how pieces control squares. The king move signifies more than just an effecient way to play. It signifies that the king holds just as much presences because he controls 9 squares, the closest one to him in each direction. If you control that with your king, than your opponent can't move his king there. Though this may be a "so what" kind of point, keep in mind there are many people who play on chess.com who don't understand how to play pawn endgames. This idea, that a king can take away the mobility of an opponents king, plays a huge factor in the sucess, or more often, failure, of one player to correctly respond to his opponents end game tactics. Therefore, it is a very good variation to put the king move in.

(for the record, the king move still technically can lead to a 6 move checkmate. The correct, well, most correct for black, 4th move, is 4. ... gxf2. This leads to 5. Bd5    e4 (black moves his pawn to block) 6. Bxe4# . Also, I appologize if any of this has been posted already, as I haven't gotten the chance to look at all the comments.)

In this alternate line , .... does not 5. Re3 also lead to mate in 5 ?

Couldn't you get mate in 4 with Bd5 instead of king e1?

En passant - Google "Laws of Chess". Briefly, when a pawn advances two squares (and only then), an enemy pawn that could have taken it on the square it passed over (and no other unit) is allowed to do so in response (and only then).

how is fxg3 possible?