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1. Rxg7 Kh8 [ … Bxg7 2. Re8+ Bf8 3. Qg3+ etc]
2. Rg8+ Kxg8
3. Qg3+ Kh8
4. Qg7+ Bxg7
5. Re8+ Bf8
6. Rxf8#
1.Rxg7+ Bxg7 2.Qg3 Bg4 3.Qxg4 ~ 4.Qxg7#
1 rg7+ kh8
2 rg8+ kg8
3 qg3+ bg4 (stops one mate)
4 qg4+ kh8
5 rg1
or
1 rg7+ kh8
2 rh7+ kh7 ( … kg8 3 rg1+ etc)
3 qh4+ kg6
4 qg5+ kh7
5 qh5+ kg8
6 rg1+ bg4
7 rg4+ bg7
8 rg7+ kf8
9 qh8++
1.Rxg7+ Kh8 2.Rxh7+
( I was attracted by 2.Re8 pinning the B and trying for mate at g7. The best defence I could get was by 2… Bf5. So 2.Re8 Bf5 3.Rxf8+ Rxf8 4.Qg3 Bg6 5.Rxh7+ Kxh7 6.Qg5 followed by 7.Qh6+ and 8.Qg7#)
2…Kxh7 3.Qh4+ Kg6 4.Rg1+ Bg4 5.Rxg4+ Kf5 6.Qg5#
1.Rxg7+ Kh8 2.Rxh7+
(2.Qg3 h6 3.Rg8+ Kh7 4.Rxf8 Bg4 5.Qxg4 Rxf8 6.Qg7#)
2…Kxh7 3.Qh4+ Kg6 4.Rg1+
(4.Qg5+ Kh7 5.Rg1 Bg4 6.Rxg4 and mate next move. )
4…Bg4 5.Rxg4+ Kf5 6.Qg5#
Perhaps variation with single motive
1.Rxg7+ Kh8
2.Qg3 (threatening 3.Rg8#)Bg4
3.Qxg4 h6
4.Rg1 Be7
(5.Rxc4 Rh7+ 6.Kxh7 Qg8#)
5.Rg8+ Rxg8 (or 5.Rh7+ Kxh7 6.Qg7#)
6.Qxg8#
After posting the above comment, closing the computer and going to bed suddenly I felt my reply had a bug viz. 3….. h5 4.Rg1 Bg4 but today morning at the computer I found it is not so. There is saving grace 3.h5 4.Qg5 and the life is shorter.
Why do I write rot again and again? What i meant was
1.Rxg7+ Kh8 2.Qg3 h5 3.Qg5 with easy win.