Suppose that a team of $ n $ players (numbered from 1 to $ n $) sit around a circular table and there are their numbers on their T-shirts.
Let $ a_1,a_2,..., a_n $ be the sequence of numbers around the table and so $ a_n $ sits beside $ a_1$. What is the maximum value of $|a_1-a_n|+\sum_{k=1}^{n-1 }|a_{k+1}-a_k|$?
As we checked the maximum value is equal to $ n^2/2$ for even $ n $ but we can not prove it. Could you help us? Thanks in advance