Let $d_n$ be the smallest number such that $n$ disks of radius $d_n$ cover a $1 \times 1$ square. I wonder about the asymptotics of this sequence.
I was able to prove quite easily that:
$$\frac{1}{\sqrt{n\pi}}\leq d_n \leq \frac{1}{\lfloor\sqrt n\rfloor \sqrt 2}$$
The lower bound is obtained simply by observing that $n$ disjoint disks of radius $\frac{1}{\sqrt{n\pi}}$ have a total area of $1$. The upper bound is obtained by noticing that a $1 \times 1$ square can be divided into a grid of $\frac 1n \times \frac 1n$ squares, and is covered by their circumcircles.
Of course, those are very rudimentary bounds.
I wonder: is the value of $\lim_{n \rightarrow \infty} d_n \sqrt n$ known?
