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Black to move and mate in 4.
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Which mate occurs depends on where White moves his King after Black places him in check by playing 1. . . . Bd3+. If White plays
2. Ka1? Black responds with Nb3+, and White has no move other than
3. a2xN, whereupon a4xb3 discovers check and mate, i.e., mate in 3.
If White plays 2. Kc1, Black responds with Qc4+. Now White has 2 choices. If
3. Kd1, Black mates with Qc2+, 4. Ke1 Qe2++. If instead
3. Kd2, Black again plays Qc2+, and White has two last choices: If
4. Ke3, then Qe2++ is mate. And if 4. Ke1, then the same move is again mate.
Note that since White’s King in this problem stays on the dark squares throughout the sequence of moves, his Queen has no chance of rescuing him.
1 b to e6+
K to h8
N to g6+
P to g6
P h5 to g6#
2 b to e6+
K to f8
Q to f5+
K to e8
Q to f7+
K to d8
Q to d7#