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What are the minimum sets of pieces required for checkmate when the opponent has no other piece that the king?
Assuming the lone King is attempting to avoid checkmate.
As simple as a Knight or even a pawn, if you consider the fact that your opponent's pieces can block the escape squares too.
W: Kf1, Nf2
B: Kh1, h2.
The Assume that white played Nf2# in the last move. So, white has just used one knight to checkmate black king.
There are many answers to your question,
1) A rook
2) Any 2 minor pieces (Minor pieces: Bishop, Night)
3) A Queen
4) 1 pawn :)
Required, but never shown