I'm fond of the little problem posed in the beginning of [this Quanta article][1]: take $n$ generic points on a circle and draw the complete graph between those points. Into how many regions do the edges cut the circle?[![Image from Quanta magazine][2]][2]

You can compute a few examples and are quickly led to conjecture that $n$ points yield $2^{n-1}$ regions. But the next example falls short: $n=6$ points yield $31$ regions. This is a nice lesson in the surprises that math has to offer. And the actual solution to the problem can be found using a bit of combinatorial reasoning and Euler's formula, which shows how you can rope in different areas of math to solve an apparently simple problem.

  [1]: https://www.quantamagazine.org/where-proof-evidence-and-imagination-intersect-in-math-20190314/
  [2]: https://i.sstatic.net/CjaBC.jpg