Suppose we have a board $100 \times 100$ and place $100$ queens such that none attack another. Prove that each of the four $50 \times 50$ sub-boards (gotten by dividing the board in $4$) contains at least one queen. Additionally, prove that if one of the boards contains exactly one queen, replacing queens by knights would lead to knights attacking each other.
My approach has been to use that each of the row and column must contain only one queen, then utilize pigeonhole principle somehow. Still, I'm not sure how is it possible to place only one queen on $50 \times 50$ board.