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Alternative puzzle solution

Are you sure you linked the right puzzle?

That notation doesn't make sense given that puzzle.

Also, the notation as stands has the king going to g8 after a rook check on h8, which would be illegal, and then the king being taken, which would also be illegal.

I'm guessing you meant a different puzzle, and some piece other than a king for black's move :)
You can play your moves on the board against the computer clicking on the "two swords" icon.
In this case, after 1. Rh8?? start advancing the pawns...
maybe mate in 20 moves.
Yes, fuck, it's a different puzzle, so sorry about that...
Thanks anyway for the responses.

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