What is the minimum number of points randomly chosen in a $4\times4$ square (each point can be at the square's boundary) so that there is always a pair of $2$ points not more than $\sqrt2$ units apart?
Let the number of points be $n$. We can divide the square into $16$ unit squares. If $n\ge17$, by the pigeonhole principle, then at least one unit square has multiple points, and the distance of those points is not more than $\sqrt2$. If $n=16$, it can happen that no unit square has multiple points. But, I have not been able to construct an example where every $2$ points are more than $\sqrt2$ units apart.
How to construct an example of the distribution of $16$ points in the large square such that every $2$ points are more than $\sqrt2$ units apart? If such an example cannot be constructed, why, and what is the maximum number of points such that it is possible that every $2$ points are more than $\sqrt2$ units apart?
My attempt:
Put $4$ points near the bottom, horizontally $1\frac13$ units apart, as in this picture:
But I don't know how to continue.