What are fun mathematical facts for non-mathematicians?

Question

I like to spend my life with mathematics. I think it is the best thing I can do in my life. However, I have great difficulty explaining what I am doing to non-mathematicians, even educated ones. For example, my mom and dad know nothing about what I am doing even if I try to explain the philosophy and the mechanism behind it. But I want to make them feel what kind of abstract works mathematicians do by some real-life applications of mathematics such as Brouwer fixed point theorem which says, after having some certain assumption about our physical world, that if you stir a cup of tea, then after you are done there must be at least one particle in the cup that is in the precise same spot as it was before.

Can you please share some mathematical theorems, facts, and applications that would give some idea about what kind of works are done in mathematics and that are easy to state and to be understood by non-mathematicians?

Answer 1

You know the Youtube channel Numberphile? Many interviews with professional mathematicians where they introduce their favorite topics to the general public. There's even one trying to explain the topological proof of the Fundamental Theorem of Algebra, if you can imagine it.

https://www.youtube.com/watch?v=shEk8sz1oOw

EDIT: I'm sure other people here can give dozens of examples. Let me give you one which I didn't invent but which I remember quite well. Imagine you were in a courtroom facing charges of dangerous driving. Would it be enough for the prosecutor to say that the speed limit on the whole road was 100 km/h but two different CCTV cameras showed that you completed a 150-km trip within one hour?

Answer 2

As a non-mathematician, the theorem I find the most counterintuitive, interesting, and beautiful in mathematics is the one that says "there are as many points in the unit square as there are points in the unit interval." More precisely, let $S = (0,1)\times(0,1)$ be the interior of a unit square and let $I = (0,1)$ be the interior of a unit interval. Then $S$ has the same cardinality as $I$. It's quite counterintuitive, because the set of points in the interior of a two-dimensional unit square should have much larger cardinality than a set of points in the interior of a one-dimensional unit interval. But it is precisely this counter-intuition that makes the theorem interesting and beautiful to me.