Proof the next problem by using the pigeon hole principle

Question

There are $2$ out of each $6$ points in a rectangle of $3\times 4$ with a distance $\le \sqrt 5$.

I know you need to divide the rectangle into $5$ equal parts so that in one part you'll get at least 2 points by the pigeon hole principle.

And I thought making rectangles of $2\times1$ in the entire rectangle but then you'd divide the rectangle in six parts.

Can somebody give me a hint on how to proceed?

Answer 1

Partition the rectangle $[0,4]\times[0,3]$ into $5$ regions by using the following 7 segments: $$(0,1)(1,2),\;(1,2)(1,3),\;(1,2)(2,1),\;(2,1)(2,0),\;\;(2,1)(3,2),\;(3,2)(3,3),\;(3,2)(4,1).$$ The diameter of each region is $\sqrt{2^2+1^2}=\sqrt{5}$.