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Well, winning a piece comes to mind. 1. Be4+ Bxe4 (if K moves, White just captures the bishop on c2 and still has the threat of Ne4+) 2. Qh5+ Kf6 3. Nxe4+ followed by capturing the bishop on d6. Perhaps there’s something stronger, but this looks like a very practical solution
1. Be4+ Bxe4
2. Qh5+ Kf6
3. Nxe4+ wins the d6 Bishop
3. Rxd6+ Ke7 [ … Qxd6 4. Nxe4+ forking the queen ]
4. Qf7+ Kxd6
5. Nxe4# (I was going to say the queen falls, but then it seems to be mate!)
Mate in 3
Rxd6+ Qxd6
Qh5+ Kf6
Qf7 mate
Sorry, I overlooked …….Kxg5
1, Rxd6+ Qxd6
2. Qh5+ Kf6
3. Qf7+ Kxg5
4. h4+ if …Kxh4 Qh5 mate or if …Kxg4 ??????
I think the two moves 1.Be4 and 1.Rd6 are roughly equivalent, and maybe exactly equivalent through transposition. Just to avoid mate, black will have to give up the queen at some point after white has sacrificed the exchange at d6 in either line.
1. Be4+ Bxe4
2. Rxd6+ Qxd6 (forced)
3. Nxe4 and there’s no time to save the black Q, ’cause of the threat 4. Qh5#
and 1. Be4+ Kf6 is also hopeless