Question 54
Let $a_1, a_2, \dots, a_n$ be an arbitrary arrangement of numbers $1, 2, \dots, n$ on a circle. Find
$$
\min \sum_{j=1}^{n} |a_j - a_{j+1}| \quad \text{and} \quad \max \sum_{j=1}^{n} |a_j - a_{j+1}|,
$$
where $a_{n+1} = a_1$ and the extrema are taken over all possible arrangements of $1, 2, \dots, n$.
$$
\min \sum_{j=1}^{n} |a_j - a_{j+1}| \quad \text{and} \quad \max \sum_{j=1}^{n} |a_j - a_{j+1}|,
$$
where $a_{n+1} = a_1$ and the extrema are taken over all possible arrangements of $1, 2, \dots, n$.