3 of 5 added 44 characters in body

Two possibilities for a four-move mate:

  1. Kd7

  2. Bb8 OR Bxc5

  3. Kc7

  4. d6

Proof that this is optimal:

  • The only white piece which can guard the square a8 is the white-square bishop: obviously the black-square bishop can't, the king can't get close enough, and the pawn can't promote in time.

  • So in order to get a checkmate, the pawn has to move out of the white-square bishop's path.

  • In order for the pawn to have space to move, the black-square bishop must get out of its way.

  • In order for the black-square bishop to be able to move, it must first be unpinned, so the king has to move.

So we need at least one move from each of the pawn, the black-square bishop, and the king.

  • The only piece which can place the black king in check is the black-square bishop. Regardless of whether it checks from c5 or from b8, the white king must guard the square b8, and he needs at least two moves to get near enough.

So the king must make at least two moves.

Thus, four moves is the minimum possible.