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I think White has a mate in 4, starting with:
1. Rg4!
For example:
1. … h6 or h5 or Bf5
2. Bh6+ followed by
3. Qg7 #
1. … QxQ
2. BxQ+ BxR
3. RxB #
1. … BxR
2. RxB QxQ
3. BxQ #
1. … BxR
2. RxB h6 or h5
3. Bh6+ followed by
4. Qg7 #
1. Rg4 threatens mate by moving the bishop, and black cannot really do anything. BxR is met by RxB which repeats the threat, QxQ is met by BxQ which creates the same mating pattern and h5 is met by Bh6+ which allows Qg7# next turn.
or… if Black uses the N in an effort to block:
1. Rg4, Bxg4
2. Rxg4, Ne4
3. Bh6+, Ng5
4. Qg7#
the same result is acheived as already indicated – mate in four!
Dr. O