knight to e8 followed by bishop to g7 mate no mater wat black plays

Ne8? Nxf8 and no mate

hxg6 is the first move.

daivydan, Ne8 fails to Nxf8, now 2.Bg7 is not mate, Kh7

johnkorean is right.  You have to go regression analysis.  It is physically impossible for Black to have moved his Knight or Bishop on the last move.  In theory, he could have moved his King out of check from g7 to h8, but for that to happen, White's last move would have had to have been Bh6+, but how could that have possibly gotten there?  It couldn't.  Therefore, Black's last move was not Kg7-h8.  Of course the f-pawn didn't move.  That leaves the g-pawn.  There is no way the g-pawn went from g6 to g5 as then the White King would have been in check with Black to move.  It is also impossible for Black to have made a capture on g5 as f6 and h6 are occupied by White pieces.

THEREFORE - Black's last move had to have been g7-g5, making 1.hxg6 legal, and in this case, CORRECT!  If Black makes any of the 3 legal knight moves, 2.g7 is mate.  If Black takes the pawn with 1...fxg6, then 2.Nxg6 is mate!