Your king is on a corner cell of a chessboard and your opponent's knight is on the corner cell diagonally opposite. No other pieces are on the board. The knight moves first.
For how many moves can you avoid being checked?
Source: David L. Silverman's collection of game problems, titled Your Move.
(In reply to Solution
The king does not have to be close to the center of the board to be safe.
If he is on a corner, then there are only two squares of the opposite color to which he can move. If the knight is attacking both of those squares, then the king is forced to stay on the same color, and can be checked after he moves.
If he is on an edge that touches a corner, then the king has three squares of opposite color available, and the knight can attack at most two. So a king that is on an edge can always move to an opposite color. If the only available square of the opposite color is the corner, then the King can safely move there, because if the knight was blocking the other two squares, then the knight is not in a position to trap the king in the corner, and the king will be able to move to an opposite color on the next turn.
IN SHORT, THE KING CAN AVOID BEING CHECKED IMMEDIATELY IF HE ALWAYS MOVES ONTO A SQUARE OF THE OPPOSITE COLOR FROM THE KNIGHT. HE CAN AVOID BEING CHECKED INDEFINITELY IF HE ONLY MOVES TO A CORNER WHEN THAT IS THE ONLY AVAILABLE SQUARE OF THE OPPOSITE COLOR THAT IS NOT UNDER ATTACK BY THE NIGHT.
Following those rules, the King is safe anywhere on the board, not just close to the center.