NL-NWO 2000 Problem 5

Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules: (1) from the square numbered " $n$ " to the square numbered " $2n$ ", and vice versa; (2) from the square numbered " $n$ " to the square numbered " $3n + 1$ ", and vice versa. Show that the pawn can reach the square numbered " $1$ " in a finite number of moves.