There are five points in a square with sides of length one. Prove that there are two of these points having distance at most $\frac1{\sqrt {2}}$.

Question

how would I prove this exercise:

If we had five points in a square with sides of length one. How can we use the Pigeonhole Principle to prove that there are two of these points having distance at most $\frac1{\sqrt {2}}$

Answer 1

Divide the square into quarters. Let 4 points be at the corners of the original larger square. Now, clearly the maximum distance is the line from a corner of the larger square to its centre. This distance is $\frac{1}{\sqrt2}$ units. Therefore, there are at least 2 points with distance $d$, such that

$0<d\leq\frac{1}{\sqrt2} \square$

Answer 2

Other posters assume (without proof) that a maximal configuration will have four of the points at the corners of the square. But this is not necessary.

Divide the square into four smaller squares in the obvious way. Then, by the pigeonhole principle, there must be a square that contains at least two of the five points. The diagonal of each smaller square is $\dfrac{1}{\sqrt 2}$, so the distance between these two points is at most $\dfrac{1}{\sqrt 2}$.