I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also like to keep the whole thing to 15 minutes or less.

But here is the hitch: The competition was divided into age groups. The youngest are about nine years old; the oldest are college students. I'll be speaking to the winners in all age groups at once.

Apparently a speaker in a previous year found a way to talk a little about the Gauss-Bonnet theorem to this diverse crowd. I don't know what that way was.

I've thought about the following:

- A few examples of apparently "pure" mathematics that turned out to have important applications. Graph theory informs the design of printed circuits. Hilbert's program to prove the consistency of mathematics led to the need for a precise definition of "proof", which led to Turing machines, which led to the existence of universal Turing machines, which eventually informed the design of computers. Of course there's also elliptic curve cryptography....
- A few words on the theme "mathematics is the only subject that stands on its own" in the sense that to really understand psychology, you have to learn some biology; to really understand biology, you have to learn some chemistry; to really understand chemistry, you have to learn some physics; to really understand physics, you have to learn some math, but to really understand math all you need to think about is math. (I think I will
**not**pause to acknowledge and refute those who say that to really understand math you need to really understand philosophy....). And a few words about why this is a really cool thing about math. - Just some words on math as a lifelong adventure, something you can think about whenever and wherever you are, something you can share with people of all cultures and backgrounds, and wishing them a bon voyage as they set off on this journey.

Any comments on the above, or any alternative suggestions?

**Edited to add:** I'm grateful for the many answers. In some cases the posters seemed to me to be overly optimistic about what might hold the attention of a nine year old. Here is the talk I ended up giving.

definitelynot a career I knew could exist, and the adult me would now also like to hear this hypothetical talk. $\endgroup$10more comments