Question 110
Permutation $(a_1, a_2, \dots, a_n)$ of the set $\{1, 2, \dots, n\}$ is called **messy** if for any three indices $1 \le i \lt j \lt k \le n$, $a_i + a_k - 2a_j$ is not divisible by $n$. Find all natural numbers $n \ge 3$ for which there exists a messy permutation of the set $\{1, 2, \dots, n\}$.