Question 167
For a positive integer $n$, consider $a_1, a_2, \dots, a_{2n}$ as a permutation of the first $2n$ positive integers. A permutation is called "nice" if $a_i + a_{n+i} = 2n+1$, $i = 1, 2, 3, \dots, n$ and $a_i - a_{i+1}$ is not congruent to $a_j - a_{j+1}$ modulo $2n+1$ for all $1 \le i \lt j \le 2n$ (suppose that $a_{2n+1} = a_1$).
1. For $n = 6$, give an example of the nice permutation.
2. Prove that for all $n$, there always exist some nice permutation.
1. For $n = 6$, give an example of the nice permutation.
2. Prove that for all $n$, there always exist some nice permutation.