BY-MO 2001 Problem 6

Let $n$ be a positive integer. Each square of a $(2n-1) \times (2n - 1)$ square board contains an arrow, either pointing up, down,to the left, or to the right. A beetle sits in one of the cells. Each year it creeps from one square in the direction of the arrow in that square, either reaching another square or leaving the board. Each time the beetle moves, the arrow in the square it leaves turns $\frac{\pi}{2}$ clockwise. Prove that the beetle leaves the board in at most $2^{3n-1}(n-1)!-4$ years after it first moves.