Question 6
For each positive integer $n$, consider a permutation $a_1,a_2,\ldots,a_{2n}$ of the first $2n$ positive integers. Call it beautiful if opposite entries satisfy $a_i+a_{n+i}=2n+1$ for $1\le i\le n$, and the $2n$ differences $a_i-a_{i+1}$ are pairwise incongruent modulo $2n+1$, with $a_{2n+1}=a_1$.
(a) Give a beautiful permutation for $n=6$.
(b) Prove that a beautiful permutation exists for every positive integer $n$.
(a) Give a beautiful permutation for $n=6$.
(b) Prove that a beautiful permutation exists for every positive integer $n$.