I am running a small chess school for beginners. During a class I asked a child to compose a checkmate position using only king, bishop and knight against the opposing king in a corner. Then I made the task tougher by placing the opposing king in the middle edge. And finally I asked him on purpose to try the same in the center of the board, to see if he was able to conclude that the task was impossible. Not only he did, but he also proposed to try the task in the center of the board with a third minor piece. To my surprise, this task seems to be impossible as well. We tried with any combination of three minor pieces plus the king. Is it true that this task is not possible?
It is indeed impossible to construct a mating net with king + 3 minor pieces against a lone king in the center of the board.
A mating net would require that all 9 squares in the 3x3 block which contains the opposing king be under attack. The king can only cover 3 of those 9 squares since it can't be next to the opposing king. This would leave a 3x2 rectangle of squares which would need to be covered by the 3 minor pieces. That rectangle has 3 black and 3 white squares. Any minor piece can cover 2 of those 6 squares -- but those squares would need to be of the same color. (One might quibble that a bishop which is directly adjacent to the opposing king could cover 3 squares, but that would be capturable by the king unless supported by another piece, the net result being just 3 squares covered by 2 pieces). If 1 of the minor pieces covers 2 of the white squares and 1 covers 2 of the black squares, the remaining piece would have to cover 1 white and 1 black square, but that is impossible.