Distribution of points and Pigeon-Hole Principle I found this practice problem in a textbook...
"Ten points are given within a square of unit size. Then
there are two of them that are closer to each other than 0.48, and there are
three of them that can be covered by a disk of radius 0.5."
The solution in the book assumes that these 10 points are placed evenly on the area, as if one had divided the square into 9 "sub-squares" and had placed 1 point in each of these "sub-squares".
To me it's not obvious that one would make such an assumption. On the other hand the problem doesn't specify any particular arrangement of these 10 points. Since I lack mathematical maturity I was wondering you opinion on the "clarity" of this problem and maybe some advice on how to deal with such (perceived) ambiguity.
 A: The problem does not assume that the points are placed evenly.
Instead, here is is going on:
The problem first takes the unit square and divides the unit square evenly into 9 smaller squares. Note that every single point in the unit square is in at least one of the 9 smaller squares (possibly more than one, if it's on a boundary).
Label the 9 smaller squares from 1 to 9.
Now, for each of the 10 points, this point is inside the unit square and therefore inside at least one of the 9 smaller squares. So for each point $p$, find some $n$ between $1$ and $9$ such that the point $p$ is in the square labelled with number $n$. Then, label that point $p$ with the number $n$.
Now there are 10 points. And there are only 9 possible labels. This means that we can find two distinct points that have the same label (this is the pidgeonhole principle).
Take points $p_1$ and $p_2$ with the same label. Then it must be the case that $p_1$ and $p_2$ are both within a square of side length $1/3$, so their distance cannot exceed $\frac{\sqrt{2}}{3} < 0.48$.
