1. Let $a_{1}, a_{2}, \cdots, a_{17}$ be a permutation of $1,2, \cdots, 17$, and satisfy
$$
\begin{array}{l}
\left(a_{1}-a_{2}\right)\left(a_{2}-a_{3}\right) \cdots\left(a_{16}-a_{17}\right)\left(a_{17}-a_{1}\right) \\
=2^{n} .
\end{array}
$$

Find the maximum value of the positive integer $n$. (Supplied by He Yijie)
1. From part $(3)$ of the 4th question of Grade 10, we know $n \leqslant 38$.

Below, we construct a permutation $a_{1}, a_{2}, \cdots, a_{17}$ of $1,2, \cdots, 17$ such that $S=2^{38}$.

For example, let $a_{1}, a_{2}, \cdots, a_{17}$ be $1,17,9$, $5,13,11,3,7,15,16,8,12,4,6,14,10,2$. In this case, $S=(-16) \times 8 \times 4 \times(-8) \times 2 \times 8 \times$ $(-4)(-8)(-1) \times 8 \times(-4) \times 8 \times$ $(-2)(-8) \times 4 \times 8 \times 1$ $=2^{38}$.
In conclusion, the maximum value of $n$ is 38.