I once attended a talk in which nothing but the classification of Platonic solids was explained to a general audience in order to show what math is like.

The speaker was doing really great in drawing beautiful pictures at the blackboard. Very little notation was used and no slides where involved. Even if you didn't manage to follow his arguments to the very end you could read off his fascination of his face and look at the pretty pictures.

He gave the talk in such a way that not a finished proof was presented but the object where explored together with the audience and steps of the proof where found in a live-discussion.

Maybe you can tell from what I am writing that this talk was really impressive and reached it's goal to teach that math can be a beautiful and fascinating way of exploring structures (or solving puzzles if you wish to say so).