Question 6
A positive integer $n \gt 1$ is called good if there exists some permutation of the numbers $1, 2, 3, \dots, n$, denoted by $(a_1, a_2, a_3, \dots, a_n)$ such that $a_i$ and $a_{i+1}$ have different parities for every $1 \le i \le n-1$; and for every $1 \le k \le n$, the sum $a_1 + a_2 + \dots + a_k$ is a quadratic residue modulo $n$. Prove that there exist infinitely many good numbers, as well as infinitely many numbers which are not good.
*Remark: Here an integer $x$ is considered a quadratic residue modulo $n$ if there exists an integer $y$ such that $x \equiv y^2 \pmod{n}$.*
*Remark: Here an integer $x$ is considered a quadratic residue modulo $n$ if there exists an integer $y$ such that $x \equiv y^2 \pmod{n}$.*