Is there a great mathematical example for a 12-year-old? I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets.

Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and understanding some of the maths they did - the real thing.

So what else could we have done - thinking that we know our multiplication tables and fractions, but aren't yet completely confident with equations which have letters for unknown numbers?

I did think of proving that there are infinitely many primes - we can follow an argument - other suggestions welcome.

And incidentally, tell your local high school to do this ...
Six people at a dinner party is sufficient to ensure that there are either three mutual strangers or three mutual acquaintances. In fact, six is the smallest number that ensures this phenomenon. This is the diagonal Ramsey number $R(3,3)$, and the proof can be demonstrated with a couple pictures and just a dash of the pigeonhole principle. There are lots of directions she could go after understanding $R(3,3)$ (though most of it is not due to Ramsey).
Show her that 2 + 2 may not equal 4, but for example may equal 12, and explore the implications of that fact with her! Subtracting from each side 2 and then 2 again you'd get 0 = 8, and this would be a great intro to modular arithmetic. 


Moreover, this is a great example of a situation in which dismissing what's known and allowing a seemingly wrong answer can open new doors. And it is also a way of showing that, in maths, there are no wrong ways (unless you contradict yourself, of course), but only roads that may lead nowhere (but that are worth exploring though). 


An intro to quotients with simple and concrete examples would be mind-blowing as well, I believe. Let her come up with her own answer to what happens if, on the [0, 1] segment, we make 1 and 0 be the same element (we get a circle!). Then you can go on with more difficult examples and build a cylinder, a torus and a mobius string from a unit square! Funny enough, modular arithmetic is based upon quotients! 
Show that the commutative property may not hold in infinite cases. Take the $\sum_{i=0}^\infty(-1)^n$ and ask her what she thinks it may sum up to. That sum waves up and down never approaching anything, but rearranging its terms it can be made divergent.


This is a great example of how things may not always be true in more difficult cases, even if they are in simpler ones.