PL-OM 2003 Problem 6

Let $n$ be an even positive integer. Show that there exists a permutation $(x_1, x_2, \ldots, x_n)$ of the set $\{1, 2, \ldots, n\}$ , such that for each $i \in \{1, 2, \ldots, n\}, x_{i+1}$ is one of the numbers $2x_i, 2x_{i}-1, 2x_i - n, 2x_i - n - 1$ , where $x_{n+1} = x_1.$