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Yes indeed, it is not that easy. But once you'll find the idea it will be no longer that difficult ;-) Keep on trying :-) I will go to sleep now and see tomorrow if you (or others) have found the solution.
Edit: Yes, taking on g7 is not good, you are right.

I found the solution in six moves:

  1. Qe4 c4

  2. f4 c3

  3. f5 gxf5

  4. gxf5 Rxg5

    4. ... Rg6
    5. Qh4+ Rh6
    6. Qxh6#
    
  5. Qh4+ Rh5

  6. Qxh5#

True, that seems to work too. Very good! Now you have to improve your solution by one move :-) Edit: If I remember correctly the solution you found was by coincedence the one I found as well when I saw the study the first time. But it is not the right one. There is a better and prettier one.

I give up! I spent the whole morning trying to find the solution in five moves and I couldn't :(. This is the most difficult position I have seem, amazing...

Thanks a lot for your efforts juanmi96. I will wait a little bit before presenting the solution. Maybe one hint right away: try to understand the function of the pawn on h3 & try to understand the function of the knight on h1 :-)

I knew you would need me, haha. :)

Yepp! :-)

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