On an $8\times 8$ chess board, there are $32$ white pieces and $32$ black pieces, one piece in each square. If a player can change all the white pieces to black and all the black pieces to white in any row or column in a single move, then is it possible that after finitely many movies, there will be exactly one black piece left on the board?
The answer is No.
Assuming there are $k$ black pieces and $(8-k)$ white pieces in the row or column the player is operating on. Then after a move, there will be $(8-k)$ black pieces in this row or column. The change $(8-k)-k=8-2k$ is always an even number. This means that every move will change the number of black pieces by an even number. Hence, it is impossible to change the initial $32$ black pieces to just $1$ piece.