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Let’s try….1.Bf5 Bg2 2.Re6 Be4 3.Rxe4 Kb1 4.Re1#
This must start with a bishop move to clear 2nd rank planning Rh2:
1. Bh7/Bg6 Bf1 (1. … Bg2? 2. Rh2 Bf1 3. Ra2#. 1. … Bf5? 2. Rh1+ Bb1 3. Rxb1#.)
2. Rh2! Bg2 (2. … Be2/Bd3 3. Rh1+ … 4. RxB#)
3. Bd3!
Zugzwang. Black B can’t capture this diagonal now.
3. … any
4. Ra2#
In addition to mate by rook check on 1st rank,one should use Ra2+ but for that B has to Vaccate c2.
1.Bh7 Bf1
1… Bg2 2.Re6 (2.Rh2 fails because of 2… Be4)Be4 3.Rxe4 (not 3.Bxe4 stalemate)Kb1 4.Re1#
2.Rh2 Bg2
3.Bd3!
(3.Bg6 or Bf5 fail due to 3…. Be4)
3…. B moves
4.Ra2#
Other B moves by white on first move fail.
(1.Bg6 blocks 2.Re6 required after 1…. Bg2.)
(1.Be4 is met by 1…. Bg2.)
(1.Bd3 Bf1 2.Bh7 will mate in 5.)
Rook moves also fail
(1.Rf6 Bf5)
(1.Rd6 Bg4 2.Rd2 Be2 3.Bh7 Bd3 4.Rd1+ Bb1 5.Rxb1#)
There are many ways to win in this position. Stalemate is a possibility so white has to be careful. Here are two variations that mate in 4 moves.
1.Bh7, Bg2
2.Re6, Be4
3.Rxe4 (3.Bxe4 is stalemate.), Kb1
4.Re1#
Another possible black defense is:
1.Bh7, Bf1
2.Rh2, Bg2
3.Bd3 (3.Rxg2 is stalemate.), Be4
4.Ra2# (Again, 4.Bxe4 is stalemate.)
1.Bh7…
a)…Bf1 2.Rh2…(…Bd3 3.Rh1+ Bb1 4.Rxb1# ) Bg2 3.Bd3 Be4 4.Ra2#
b)…Bg2 2.Re6 Be4 3.Rxe4 Kb1 4.Re1#