Divide the squares of the chessboard into 10 sets, each set
consisting of the 10 squares labelled with the same digit in
the diagram below:
If each set contained at most 4 rooks then there would be at most
4 * 10 = 40 rooks on the board.
But 41 rooks are on the board, so at least one of the sets
must contain at least 5 rooks, and those rooks do not attack each other.