Question 3
Let be given an integer $n \ge 1$. Consider a permutation $(a_1, a_2, ..., a_{2n})$ of the first $2n$ positive integers such that the numbers $|a_{i+1} - a_i|$ $(i = 1, 2, ..., 2n-1)$ are distinct each from others.
Prove that $a_1 - a_{2n} = n$ if and only if $1 \le a_{2k} \le n$ for every $k = 1, 2, ..., n$.
Prove that $a_1 - a_{2n} = n$ if and only if $1 \le a_{2k} \le n$ for every $k = 1, 2, ..., n$.