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.
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 $\frac{\sqrt{2}}{3} < 0.48$.