How do you stalemate with king bishop vs king?
If a player still has a pawn, this pawn can be promoted to a
greater piece (Queen, Bishop, Rook, or Knight -- usually the Queen
is chosen) and, if the queen or rook is chosen, this means a win
for the player starting with the pawn. Stalemate only occurs if the
pawn is captured by the opponent before it can be promoted. Also,
if a player is left with only a king and knight, or only a king and
bishop, and the opponent only has a king, or a king with either (a
bishop or knight), in these situations, it is impossible to bring
the opponent to checkmate. Only with the king and a rook or
(obviously) with the king and a queen, can checkmate be
accomplished.