Question 38
Let $n$ be a positive integer. Consider an $n \times n$ chessboard with certain cells colored green. A rook can be placed on any green cell and moves only to other green cells, changing its direction horizontally and vertically with each subsequent move. It is important to note that remaining in the same cell is not considered a valid move. It is known that a rook cannot return to its starting cell within six moves. Prove that the number of green cells on the chessboard is less than $2n(1 + \sqrt[3]{n})$.