a) White to mate in one.
b) White to mate in two.
c) White to mate in three.
d) White to mate in six.
Solutions: a) N-f7 mate
b) N-f6 ch, K-h8 then N-f7 mate
c) 1) N-e7 ch, K-h8, 2) N-d8 (taking the pawn leads to a stalemate, as the king has no legal moves), P-g4 3) N-f7 mate
d) 1) N-e6 ch, K-g8 2) K-g6, P-c3 3) N-e4, P-c2 4) N-f6 ch, K-h8 5) N-g5, P-c8 (Q)
6) N-f7 mate! PS - Two knights and a king cannot force a checkmate against a lone king. Having that black pawn on the board is what made this mate possible. So remember, to avoid a draw, leave at least one pawn on the board! (In problem #1 above, two knights and a king will indeed be checkmating a lone king, but this position cannot be forced. Black would have to make a poor move to get itself into such a position - it cannot be forced!)