IR-TST 2002 Problem 12

We call a permutation $\left(a_1, a_2, ..., a_n\right)$ of $\left(1, 2, ..., n\right)$ quadratic if there exists at least a perfect square among the numbers $a_1$ , $a_1 + a_2$ , $...$ , $a_1 + a_2 + ... + a_n$ . Find all natural numbers $n$ such that all permutations in $S_n$ are quadratic.

Remark. $S_{n}$ denotes the $n$ -th symmetric group, the group of permutations on $n$ elements.