So close...
Chess Puzzle So Hard Garry Kasparov Couldn't Solve It
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Still, so close 
(Edit: there are apparently a lot of mate in 6's following the riddle's conditions...)
I did find a mate in 5 ending with the black knight capturing the white rook, but it doesn't start with e4. This is really difficult, I still am working on this
Here is the solution:
greekgift_221b wrote:
Well, what you should notice is that move order is the most important. If you see, like in #4 that it does not matter if the move is Rg8 or a6, you know that you are wrong.
Well yeah, that solution is wrong beforehand. It starts with 1. f3, while the task is to start with 1. e4.
greekgift_221b wrote:
That is exactly why it says that first move MUST be e4.
It was the only solution I've had with the fifth move being knight takes rook #. Again, very difficult riddle
A game begins e4 and ends on move 5 with the move knight takes rook checkmate. What was the whole game? Black is the one delivering checkmate