Show that if there are 15 points scattered inside a 1 unit square, then at least 3 of them can be covered by a square of side $1/5$

Question

I have been trying to find a proper way to divide the square into smaller squares but nothing seems to be working so far, that is, trying to apply in a straight forward way the pidgeonhole principle. I tried dividing the square into rectangles but that didn't seem to work either (14 rectangles of sides $1/7\times 1/2$). Could this be solved by contradiction? Any suggestions?

Answer 1

I'm sorry, but I simply don't see it:

or better:

Answer 2

The statement is not true. Place two points at each of the dots in the figure. The side of the inner square is $\frac {\sqrt 2}4$, greater than the diagonal of a square of side $\frac 15$. The distance from a corner to a corner of the center square is greater yet. By expanding the central square this construction works for $16$ points up to a distance of $\frac{\sqrt 3-1}{\sqrt 2} \approx 0.51764$, so we are not even close.