BG-MO 1991 Problem 6

White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.