Question 26
Let $n$ be an even positive integer. Show that there is a permutation $x_1, x_2, \ldots, x_n$ of $1, 2, \ldots, n$ such that for every $1 \leq i \leq n$ the number $x_{i+1}$ is one of $2x_i$, $2x_i - 1$, $2x_i - n$, $2x_i - n - 1$ (where we take $x_{n+1} = x_1$).