$16 \cdot 17$ Given that there are seven points $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}$ (not necessarily equally spaced) on a line, and let $P$ be any point selected on the line, and let $s$ be the sum of the lengths of the segments $P P_{1}, P P_{2}, \cdots, P P_{7}$. The position of $P$ when and only when $s$ is minimized is
(A) the midpoint of $P_{1} P_{7}$.
(B) the midpoint of $P_{2} P_{6}$.
(C) the midpoint of $P_{3} P_{5}$.
(D) $P_{4}$.
(E) $P_{1}$.
(14th American High School Mathematics Examination, 1963)
[Solution] First, consider the case with two points $P_{1}, P_{2}$. Here, $P$ can be any point on $P_{1} P_{2}$. Notice that
$$
s=P_{1} P+P P_{2}=P_{1} P_{2} .
$$

Next, consider the case with three points $P_{1}, P_{2}, P_{3}$. Clearly, to minimize $s$, we should choose $P$ to be $P_{2}$, in which case
$$
s=P_{1} P_{2}+P_{2} P_{3}=P_{1} P_{3} .
$$

Similarly, for any $n$ points on a line:
If $n=2 k$, then $P$ should lie on $P_{k} P_{k+1}$;
If $n=2 k+1$, then $P$ should coincide with $P_{k+1}$.
Thus, when $n=7$, $P$ should be chosen at $P_{4}$.
Therefore, the answer is $(D)$.
Note: This problem can be translated into an algebraic problem as follows:
If $x_{1}<x_{2}<\cdots<x_{n}$, find the minimum value of $f(x)=\sum_{i=1}^{n}\left|x-x_{i}\right|$.