Prove that for all n exist order in which will not any average

Question

Prove that for all $n \in \mathbb N $ from $1$ to $n$ exist order in which will not any average of two numbers. For example $2,3,5,1,4,6$ is incorrect because between $6$ and $2$ is their average $4$, but $4,2,6,1,5,3$ is correct because between $6$ and $2$ is not their average, or between $5$ and $1$ etc.

Answer 1

I understand that you ask the following problem:

Prove that you can order the numbers $1,2,3,.., n$ in such a way that for any two numbers in the list, their average is not between them.

Hint 1: If you can order the numbers $1$ to $n$ in some order $a_1,..,a_n$, prove that $$2a_1,.., 2a_n, 2a_1-1,....,2a_n-1$$

is a good ordering for the numbers between $1$ and $2n$.

Hint 2: If you can find a good ordering for the numbers between $1$ and $2n$, how can you find a good ordering for the numbers between $1$ and $n+1$?