The social distancing He was the protagonist of our lives during the pandemic. But its application goes well beyond an emergency period. For example, you have ever come to enter the lift with strangers and try to put the as far as possible From everyone? Or maybe at the restaurant for a romantic dinner would you like your table to be far from all the others?
It may seem like an easy problem to solve but in reality it is very difficult that it has not yet been completely solved. In this article we see some of the solutions that offers us the mathematics For the simplified case of square -shaped rooms.
Social distancing: the easiest cases
It looks like a trivial problem, but trying to disappear In a space in order to be the as far as possible One from the other is a very complex problem. But let’s go to degrees.
The most case easy of all is when, in one square room, People are alone two as it is enough that they put themselves in two opposite corners. Similarly, if the people are four, the solution is that which would think of everyone: in fact, it is enough that each person occupies one of the four corners to have the optimal disposition. With other numbers, however, this is complicated so much that, from a mathematical point of view, it is a open problem, That is, a problem for which there is no general solution.
The case of the 3 people and how the mathematicians see it
We begin to raise the bar a little and bring the number of people to three. Already with three people things are not so simple: the best arrangement, in fact, is not obtained by occupying the corners but forming a Stortto equilateral triangle As we can see in the figure below.
In the figure above, we can see that each person is “surrounded” by one circular area gray. It deals with a sort of “safe areas“: The idea is to pretend that each person wants, around him, one of his personal area free from interactions with others, in practice a centered circle right where the person is located. This is precisely the way mathematicians look at this problem, pretending that each person is a point with a circle around: the bigger the circles and the more people are far away from each other.
In other words, our problem of people who want to stay away in a room is equivalent to the mathematical problem that consists in looking for:
Which is the square smaller in which it is possible enclose a number of rim the same?
Translated into everyday life language is like wondering:
If I have a number of cans, What is the square box smaller in which I can close them and how I will have to arrange them to occupy the least possible space.
It is a open problem for which there is no one general solution But only many Special solutionsthat is, case by case. The mathematicians have in fact calculated solutions for at least up to 10000 rims (or people!), But only for the first 30 cases have they found optimal solutions, the others can still be improved!
This means that there is no general response to the problem of finding the best way to arrange yourself in a room, being as far away as possible from each other. There are only specific solutions based on the number of people involved. Let’s see some.
Some examples of solutions to the problem of social distancing
Obviously there are trivial cases such as those of 5 peopleor of 9 people To be placed in a room, which you see represented in the figures below and that do not need explanation.
However, you don’t have to go very far to find a little less regular cases, this already happens if we have to arrange in one room 7 or 13 peopleor in the rather disordered case of 11 people That we show in the figure below, where even two people manage to have some more free space than the others.
And if instead that in a room we found ourselves in beach? Here is the best way, at least according to mathematical studies, to have 19 umbrellas on a small square beach.
