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1.Rxe8!!.Rxe8.2.Nh6+!.
>A-2….gxh6.3.Qg4+!.
>>A1-3….Kh8.4.Bf6+.Bg7.5.Qxg7#!
>>A2-3….Bg7.4.Bf6! And 5.Qxg7#!
>B-2…Kh8.3.Nxf7+.Kg8.4.Qxd5! And black is bust! He has no satisfactory defense to the double threats through double checks of mate at f7 or g8 and losing the Losing heavy material with black king still in mating net eg.
4….Ne7.( to cover f7) 5. Nh6++.Kh8.6.Qg8+!!.Nxg8. 7.Nf7#!! Or
4….Be7. To cover g8 with R.5.Nh6++.Kh8. 6.Qg8+!!.Rxg8. 7.Nf7#
4….Qc7 ( to cover f7.).5.Nh6++.Kh8.6.Qg8#!!
4….g6.5.Bf6!! And the King cannot escape!
4….h6. Will also succumb to Nxh6++ quickly or even Nd6++ followed by Nxe8 netting the R with a raging attack!
It is interesting to note that white can transpose the first two moves for a win!
1.Rxd8.Rxd8.2.Nh6+!!
1.Nh6+!!.Kh8.2.Rxd8!!
The results I believe are the same. The second is forceful as it gives B no choice. The fist carries the same threat as the first but holds the N sac for B to decide!
Rxe8-Rxe8 . Nh6-gxh6 . Qg4+ -kh8 .
Bf6+ -Bg7 . Qxg7++
Whether white should play 1.RxR RxR 2.Nh6+ or directly 1.Nh6+ looks like matter of taste.So at random
1.Nh6+ Kh8
(1…. gxh6 2.Rxe8 Rxe8 3.Qg4+ Bg7 (3… Kh8 5.Bf6+ Bg7 6.Qxg7#) 4.Bf6 followed by 5.Qxg7#)
2.Nxf7+ Kg8
3.Qxd5 threatening smothered mate
3…. h6
4.Nd8+ Kh8
5.Rxe8 and wins.
1. Nh6+ gh
2. Re8 Re8
3. Qg4+ Bg7
4. Bf6 and black cannot avoid mate at g7.
1. Nh6+
If 1…gh 2. R:e8 threatens mate beginning with 3. Qg4+ so black must allow the other R to fall.
If 1…Kh8 2. N:f7+ Kg8 3. Q:d5 threatens mate as well.
1. Nh6+ Kh8 (if 1… gh 2. Qg4+ with Bf6 threat) 2. Nf7+ Kg8 3.Qd5! threating smoothered mate.
Nh6 wins convincingly:
1. Nh6 Kh8 (gh6 is below)
2. Nf7 Kg8
3. Qd5!
And the main threat is Nh6 with double check followed by Qg8#. Black can’t guard g8 by moving the bishop since white will just play Qg8+ followed by the smothered mate of Nf7. The only defense I see from this point is to clear the h6 square for the black king, but this won’t save the position:
3. ……h6
4. Nd6 and black loses major material, and white may have even better moves at move 4 than Nd6- this is just the obvious way to win to my eye and the one I would certainly continue with.
Oh, I forgot to follow up on black’s continuation at move 1:
1. Nh6 gh6
2. Re8 Re8 (or just concede material)
3. Qg4 Bg7 (Kh8 4.Bf6 Bg7 5.Qg7#)
4. Bf6 and black can’t prevent mate, only delay it.