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Simply has to be c6. 1.Bb7 fails because the rook at d8 is guarded by the black bishop, so white needs two threats against the b7 pawn. In addition, if black passes the pawn with b6, white can sacrifice at c8 and mate with c7 with a discovered check:
1. c6 Rb4 (the only delay!!!)
Or [1. …bc6 2.Bc6 Bb7 3.Qb7#]; or [1. …b6 2.Qc8! Rc8 3.c7#]; or [1. …a6/a5 2.cb7 Ka7 3.bc8Q#]; and, of course, black cannot capture at d5 without allowing Qc8#. Continuing:
2. ab4 and mate in 2 more is unstoppable.
1.c6 b6 2.Qxd8! Bxd8 3. c7+ Bb7 4.c8(Q) mate.
Kamalakanta
I would move C6 as the pawn will take pawn and bishop will come in for mate.
A little more firepower needed on b7.
1. c6 b5 (or b6)
2. Qc8+ Rxc8
3. c7#
1. c6 a5 (or a6)
2. cxb7+ Ka7
3. bxc8=Q#
On 1. … a5, white can also play:
2. Qxa5+ Kb8
3. c7#
Black can only postpone this by one move:
1. c6 Rb4
2. axb4 and then the variations above apply.
1.c6 if(1….R x d5 2. Q x c8++) if (1….b x c6 2. B x c6 ++) if (1… b6 2.Q x c8 R x c8 3.c7 ++)
1.c6 if(1….R x d5 2. Q x c8++) if (1….b x c6 2. B x c6 ++) if (1… b6 2.Q x c8 R x c8 3.c7 ++)
1.c6 if(1….R x d5 2. Q x c8++) if (1….b x c6 2. B x c6 ++) if (1… b6 2.Q x c8 R x c8 3.c7 ++)
Play 1.c6 if(1….R x d5 2. Q x c8++) if (1….b x c6 2. B x c6 ++) if (1… b6 2.Q x c8 R x c8 3.c7 ++)
Play 1.c6 if(1….R x d5 2. Q x c8++) if (1….b x c6 2. B x c6 ++) if (1… b6 2.Q x c8 R x c8 3.c7 ++)
1. c6! threatening cxb7 wins.
If 1……… b6,
white finishes off with the cute
2. Qxc8+! Rxc8
3. c7 #
Xadrez,
Welcome to the blog. Note for future reference- the comments don’t post immediately if you have the correct solution, or something good enough to give it away to the other readers.