I actually think that [Hilbert's Third problem][1] is one of the explainable for school guys. It's even more cool that it exists in such a famous list close to the problems that are so tempting and not yet solved.
The question is: can one cut the cube in some polyhedral pieces, reglue them and get a regular tetrahedron? The answer is no and the theorem was proved by Dehn using so-called Dehn invariant. It uses some algebra and number theory but can be understood by high-school level guys. The time you need to explain this is 3-4 hours, so maybe it could be a little and nice course.
See, for example, Lectures on Discrete and Polyhedral Geometry by I. Pak

[1]: http://en.wikipedia.org/wiki/Hilbert's_third_problem