Baltic Way 1990 · Problem 1

Solution:

Let $a_{1} = 1, a_{2}, \ldots, a_{k} = n, a_{k+1}, \ldots, a_{n}$ be the order in which the numbers $1, 2, \ldots, n$ are written around the circle. Then the sum of moduli of the differences of neighbouring numbers is

$$\begin{aligned} & \left|1 - a_{2}\right| + \left|a_{2} - a_{3}\right| + \cdots + \left|a_{k} - n\right| + \left|n - a_{k+1}\right| + \cdots + \left|a_{n} - 1\right| \\ & \geq \left|1 - a_{2} + a_{2} - a_{3} + \cdots + a_{k} - n\right| + \left|n - a_{k+1} + \cdots + a_{n} - 1\right| \\ & = |1 - n| + |n - 1| = 2n - 2. \end{aligned}$$

This minimum is achieved if the numbers are written around the circle in increasing order.