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Ng6 (Qh8#) fe Qe6+ Rf7 Qf7+ Kh7(8) Rh3#
1. Ng6! Wins. Threat 2. Qh8#.
1……..fxg. 2. Qe6+. Kh8/7. 3.Rh3#. (If 2…..Rf7. 3. Qxf7+ and 4.Rh3#)
1……..f6. 2. Qe6+. Kh7. 3. Nxf8+.Qxf8. 4. Qxe4+. Kg8. 5. Qe6+. Kh7. 6. Rh3+ and white 2 exchanges up with a mating attack.
1. Ng6+ fg 2. Qh8+ K:h8 3. R:f8+ Kh7 4. Rh3# – much showier.
This is a very interesting position. Since posting my answer above, I have been haunted by the idea that 1…f6 isn’t the best reply.
1. Ng6 is surely the best move.
1….. f5! seems to be the best reply, it keeps White’s Q out of e6 and also keeps the f-file somewhat closed.
2. N:f8 seems to be the only option left to White since 2. Qh8+ Kf7 3. R:f5+ K:g6 and whichever way white takes the Rf8, Black plays ….Qc1+ and next takes the Re3 with check, or 3. Q:f8+ Q:f8 4. N:f8 N:e3 and Black’s mobile pawn wall becomes a huge problem for White.
2…. N:e3! otherwise White walks away with 2 Exchanges.
3. Ng6! since 3. Q:e3 Q:f8 looks like it gives Black too much play with the pawn wall in the center.
3…. Ng4 since 3….N:f1 4. Qh8+ Kf7 5. Ne5+ Kf6 6. Qe8 g4 (the only defense to 7. Qf7#) 7. Qg6+ Ke7 8. Qf7+ Kd6 9. Qd7# and White cannot be allowed to play 4. Q:f5.
4. R:f5 breaks up the pawn wall and forces the exchange of Q’s by 4….Qc1+ 5. Rf1 Qe3+ 6. Q:e3 N:e3 and white is winning.