Here's a couple of examples from the Japanese TV show "Coma University of Mathematics". There have been some excellent episodes which are accessible to non-mathematicians yet I think really conveys the creativity in mathematical research, and how very different that is from doing complicated arithmetic.
The moving sofa problem: explain that, to get to your room, you have to pass through two corridors, 1m in width, which meet at right angles. What shape of sofa has the largest area but can still get into your room? It's not immediately clear what shapes one should try, and it might surprise your audience that such a simple-looking question is an open problem.
The art gallery theorem (wikipedia is a good reference, I'm not allowed to post 2 links): in short, how many guards (or CCTV cameras) do you need to cover all of a polygonal art gallery? Again, a problem that's easy to understand, but the proof for the upper bound requires a clever idea.
