Question 6
Let $a_1,a_2,\ldots,a_{2n}$ be a permutation of $\{1,2,\ldots,2n\}$ such that
$$|a_i-a_{i+1}|\ne|a_j-a_{j+1}|$$
for all distinct indices for which these adjacent differences are defined. Prove that $a_1=a_{2n}+n$ if and only if $1\le a_{2i}\le n$ for every $i=1,2,\ldots,n$.
$$|a_i-a_{i+1}|\ne|a_j-a_{j+1}|$$
for all distinct indices for which these adjacent differences are defined. Prove that $a_1=a_{2n}+n$ if and only if $1\le a_{2i}\le n$ for every $i=1,2,\ldots,n$.