You have unlimited number of knights, bishops, rooks and kings. What is the biggest number of pieces (any combination) you can place on a chessboard, so that no piece is attacked by another one?
If we put 32 knights on all black squares, then no two pieces will attack each other. Now let's see that if we have more than 32 pieces, then there will be two which attack each other. Split the chessboard in 8 rectangular sectors of size 2x4. It is not hard to see that if we have more than 4 pieces in the same 2x4 sector, then 2 of them will attack each other. Therefore we can place at most 4x8=32 pieces on the chessboard.