$9 \cdot 43$ Find a permutation $a_{1}, a_{2}, a_{3}, \cdots, a_{1962}$ of $1,2,3, \cdots, 1962$ such that the following sum is maximized:
$$
\left|a_{1}-a_{2}\right|+\left|a_{2}-a_{3}\right|+\cdots+\left|a_{1961}-a_{1962}\right|+\left|a_{1962}-a_{1}\right| .
$$
[Solution] Let $S=|a_{1}-a_{2}|+|a_{2}-a_{3}|+\cdots+|a_{1962}-a_{1}|$. If we remove the absolute value signs, $S$ can be written as the sum of $2 \times 1962$ numbers, half of which are positive and half are negative. The absolute values of these numbers cover $1,2,3, \cdots, 1962$ and each number is taken twice. Therefore, we have
$$
S \leqslant 2(982+983+\cdots+1962)-2(1+2+\cdots+981) .
$$

Thus, the permutation of $1,2,3, \cdots, 1962$ in the following form maximizes $S$:
$$
1962,1,1961,2,1960,3, \cdots, 982,981 \text {. }
$$