$35 \cdot 50$ In a square with side length 1, there are any five points on the sides or inside the square. Let $a$ be the smallest number with the property that from these five points, one can always select a pair of points such that their distance is not greater than $a$. Then $a$ is
(A) $\frac{\sqrt{3}}{3}$.
(B) $\frac{\sqrt{2}}{2}$.
(C) $\frac{2 \sqrt{2}}{3}$.
(D) 1 .
(E) $\sqrt{2}$.
(12th American High School Mathematics Examination, 1961)
[Solution] Divide the unit square into four equal parts (as shown in the figure). In this way, among the five points within the unit square, at least two points must fall within the same smaller square (by the Pigeonhole Principle).

The maximum distance between any two points that fall within the same smaller square does not exceed the length of the diagonal of the smaller square, which is $\frac{\sqrt{2}}{2}$.
Therefore, the answer is $(B)$.