I’ve started doing an intro to number theory, and one of the cool things about it is that it examines some simple questions with complicated answers. Here’s a quick example:
Show that $$n^2 – n$$ is even, for any given integer n.
And here’s the proof: we can easily see that $$n^2 – n = n(n-1)$$. Since either $$n$$ or $$n-1$$ must be even, and the product of an even number with any other number is even, their product must also be even.
This sort of problem has a property similar to many geometry problems, which is that the original statement is easily comprehensible to people without a mathematical background. In this case, the solution also is widely accessible. There are some more complex proofs, though, and I’ve put one below the fold if you’re interested.