Let n $\ge$ 3 be an integer and let ($p_1$, $p_2$, $p_3$, $\dots$, $p_n$) be a permutation of {1, 2, 3, $\dots$ n}. For this permutation we say that $p_t$ is a turning point if 2$\le$ t $\le$ n-1 and ($p_t$ - $p_{t-1}$)($p_t$ - $p_{t+1}$) > 0 For example, for n = 8, the permutation (2, 4, 6, 7, 5, 1, 3, 8) has two turning points: $p_4$ = 7 and $p_6$ = 1. For fixed n, let q(n) denote the number of permutations of {1, 2, 3, $\dots$ n} with exactly one turning point. Find all n $\ge$ 3 for which q(n) is a perfect square.