Question 7
Two wizards perform in the following issue. At first the first wizard locks the second wizard in a cabin where he can neither see nor hear anything. To start the game, the first wizard invites Daniel, a member of the public, to place a white or black pawn on every box of a chessboard $n \times n$. Then he asks Daniele to indicate a box $C$ of his choice. At this point, the first wizard chooses a box $D$ (not necessarily different from $C$) and replaces the token found on $D$ with one of the other color (white with black or black with white). The cabin in which the second wizard was then opened is then opened. Observing the chessboard, the second wizard manages to guess what the $C$ box is. For which $n$ can the two wizards implement such a strategy that their number always succeeds?