Show that if you place 41 rooks on a 10x10 chessboard, then
you can always find 5 which do not attack each other.

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.