Minimum distance among a set of points in COVID-19 times Students going back to schools made me think of the following (non-trivial, I think) problem.

Problem: How to arrange students in a classroom so that they keep a distance of $1.5$ meters from each other in such a way that there is room for as many students as possible?

Formally, the problem consists of solving for $n$ points
$$p_0=(0,0), p_1=(x_1, y_1), \ldots p_{n-1}=(x_{n-1}, y_{n-1})$$
where $n$ is given
$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{i \neq j} \| p_i - p_j \|_2\\ \text{subject to} & \|p_i - p_j\|_2 \geq 1.5 \qquad \forall i \neq j\end{array}$$
How would you face this problem? Are there algorithms to solve it?
(I am using the Euclidean norm, so it is a non-linear problem)

The usual formulation of this problem is that you are given a fixed shape of classroom and are supposed to maximize $n$. Imagine you extended the classroom by $0.75$ metres in each direction, then you would be able to place a circle of radius $0.75$ on each student without the circles overlapping. Vice versa, any nonoverlapping packing of circles of radius $0.75$ in the extended classroom corresponds to a valid positioning of students in the original classroom.
Alternatively, your task might be to fit $n$ nonoverlapping circles of radius $r$ into a given shape, and your goal is to make $r$ as large as possible.
This type of problem is known as circle packing problems. It's popular in recreational mathematics, and I believe the usual approach to find packings is still just to try a bunch of arrangements of circles (either manually or by computer) until you get a nice one. There's no known way to verify optimality of a packing in short time.
