Math talk for all ages 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:

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*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.
 A: I have given a talk to slightly older students, but the subject might be appropriate also to 9 year old students.
The talk was about bodies of constant width. Obviously circles have the properties that they are bodies of constant width (useful if you want to place stuff on a bunch of circles aka. "wheels"). This can be demonstrated by placing a board or similar on balls and move it around. The kicker is of course that (2D) circles (or 3d balls) are not the only bodies with this property (nowadays you can find 3D models printable by a 3D printer on the internet, I think the keyword here is Meissner body). There is a lot of applications one can talk about:

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*Franz Reuleaux is said to have studied them to make buttons for his wife (I know different times) which do not roll away

*Canadian money is not round but made of shapes of constant width (some vending machines need this property to ascertain that they are handed in fact money)

*On a darker note, the challenger spaceshuttle catastrophe was at least partially caused by a "lack of roundness" (according to Feynmans memoirs) of the reusable parts which made the insulation fail. In said memoirs you find a beautiful little picture of a shape which is obviously not round but would have passed NASAs roundness test at that time (they checked roundness by measuring the width several times in certain fixed angles from each other, obviously such a test can never prove that we have constant width)

Finally, after all the hands on stuff, there are some nice mathematical theorems attached to it (e.g. Barbier's theorem 1) and even a lot of open questions when leaving 2D.
For inspiration one can look at the great book by Sagwin: How round is your circle?
They made some promotional videos 2 and have great math and engineering examples collected. This might not be exactly what you had in mind, but I had great fun showing this to the students (especially since the 3D printer people at TU Berlin made a lot of great models for my talk)
A: 
The talk can be entirely congratulatory, or it can contain a bit of actual mathematics ... any alternative suggestions?

How about some history of mathematics?
It could be structured around a few notable mathematicians throughout history (please try to include women). Or around kinds of problems that concerned people of different eras: are all numbers rational, how to solve polynomial equations, what does infinity mean, what does computable mean, ..., ending with what you feel is a central problem today.
To ensure older students haven't seen it all before, specific and obscure may be better, e.g. spending more time on anecdotes from Ramanujan's life or Erdos' than on explaining their importance.
A: Another possibility would be to talk about a few unsolved problems in mathematics.  It's easy for kids to think of math as a "finished" edifice, compared to fields like biology and physics where we hear about new discoveries frequently; seeing some unsolved problems can make it more exciting.  I think this recent book does a good job explaining some unsolved problems accessibly, and at least some of them even a 9-year-old should be able to understand.
A: I like to show how the same mathematics shows up in very different contexts. A topic that I've used with quite varied audiences (though never with as much variation in a single audience as you have) is parabolas. They show up as the paths of thrown baseballs (or fired cannonballs), as the shape of (weightless) cables of suspension bridges, as the ideal shape of radio-telescope dishes (or the reflectors at the back of automobile headlights if you want the beams to emerge parallel).  And yet, all parabolas are the same, up to scaling and orientation. You can also mention mathematical descriptions in terms of a plane section of a cone, or focus and directrix, or (if the students are OK with graphs) the graph of $y=x^2$. All these aspects of parabolas have nice pictures that you can show.
A: I've been to quite a few of such talks (though most or all in the 12-to-18-year-old range). I feel and believe that a talk that just presents something nice (to a professional mathematician) is unsatisfactory, disappointing, unfulfilling, for such audience.
If at all possible, give a talk that shows the power of math, something with punch: solve a problem. Take a real problem, a problem from reality (*) whose solution is unreachable for the audience, and solve it elegantly with math.
(*) Something you don't need math to explain, to see wherein lies the problem. Euler characteristic, or the hairy ball theorem might get you an "okaaay?, so what?". RSA public-key cryptography counts as reality, by the way (but is perhaps overused).
Some ideas:

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*Google's PageRank algorithm might (barely) fit.


*Fractals: are nice. No math punch. Unless you can show, say, that Mandelbrot set represents the set of connected Julia sets. But 9-year-old kids don't get convergence, probably? [I mean, you need to understand at least the definitions of both fractals and of connectivity to feel the punch, the bam!]


*Steiner points in the Steiner tree problem. "Find minimal path network". It's very hard to start thinking about solutions. Unfortunately I don't know Steiner point's derivation, so perhaps it cannot fit in your talk.


*Some other optimization problem, perhaps? Routing?


*If you talk about chaos (say, in the logistic map, Lorenz atractor or the weather), there's punch in math proving unpredictability [but that's subtle], but the real punch comes if math can say something in spite of chaos and unpredictability (e.g. some general property). [Nothing comes to mind there, sorry.]


*On the other hand, presenting a collection of unsolved problems may be interesting, intriguing.
Some comments on your points:

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*"Graph theory informs the design of printed circuits." Don't know what "informs" implies here exactly, but my point about punch and 'solve a problem' applies here if you just show that a circuit can be abstracted as a graph ("okaaay?, so what?").

*"mathematics is the only subject that stands on its own". It is not. You can study math on its own, certainly, but it was (and is?) born from reality. E.g. addition for counting sheep, Newton/Leibnitz analysis, and so on. It gives the why.

A: I'm fond of the little problem posed in the beginning of this Quanta article: take $n$ generic points on a circle and draw the complete graph between those points. Into how many regions do the edges cut the circle?
You can compute a few examples and are quickly led to conjecture that $n$ points yield $2^{n-1}$ regions. But the next example falls short: $n=6$ points yield $31$ regions. This is a nice lesson in the surprises that math has to offer. And the actual solution to the problem can be found using a bit of combinatorial reasoning and Euler's formula, which shows how you can rope in different areas of math to solve an apparently simple problem.
A: Following on from Wlod AA's idea, there must be plenty more juicy morsels that can be understood (if not proved!) without much knowledge.  For example:

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*The Hairy Ball Theorem (“You can't comb a hairy ball smooth.”)


*The Ham Sandwich Theorem (“However you arrange two slices of bread and one of ham, you can always cut all three in half with a single cut.”)


*The Chicken McNugget Theorem (“What's the largest number of McNuggets you cannot order without asking McDonald's to split a pack?”)
All of these are simple to understand and can be easily pictured.  Depending how much time you have, and how deep you want to go, you could simply state one or more of these (or just their names!), or you could give some examples — practical examples with props, if possible!
(You might even give a very brief sketch of how you might go about proving them, though that would probably be hard within the age-group and time limitations.)
A: My inclination would be to convey that it's fun to be a professional mathematician.
How many people in the world have a fun job that they love doing? Only a small percentage. I feel privileged to be in that group.  If you can convey that, then I think that will be a more valuable message than any message about how mathematics is useful or noble or important.  They'll hear those messages from other people.  But they may not hear very often from someone who spends all their time doing math and feels lucky to have that opportunity.
In terms of actual mathematical content, I like mathematical games and puzzles myself, so I personally would try that tack.  Kids who have won a math competition are probably going to enjoy something along those lines. But the main thing is to pick a topic that excites you personally so that your enthusiasm will be obvious.
A: I'd present a juicy morsel of mathematics, say -- the Euler characteristics theorem for $\ \mathbb S^2.\ $ I'd adopt a classic proof.
Let $\ \mathbb S^2\ $ be divided into convex geodesic polyhedra, $\ P.\ $. (Allow some neighboring edges to extend one another so that occasionally they lie on the same large circle). Then the sum of the angles of a polyhedron $\ p\in P\ $ is equal to
$$ \pi\cdot(n_p-2)\ +\ A_p $$
where $\ n_p\ $ is the number of edges (or vertices) of $\ p,\ $ and $\ A_p\ $ is the area of $\ p.\ $ Then summing over $\ p\in P\ $ gives us the Euler formula rapidly:
$$ |V| - |E| + |P| = 2 $$
where $\ V\ E\ P\ $ are the sets of vertices, edges, and polyhedra of the given scheme.
The simple combinatorial argument must be satisfying to youngsters. On the other hand, the students get a feel for the place of the general theory since they would be pointed to the measure theory. Finally, they may appreciate the power of special examples, e.g. of surfaces of constant curvature. Indeed, one can go beyond $\ \mathbb S^2.\ $ One only needs surfaces for which the sum of areas of geodesic polyhedra would be, say, $\ -8\cdot\pi\ $ (instead of $\ +4\!\cdot\pi)\ $ and everything else would be the same.

From my personal experience: I was invited to give a talk at a minor university (spring of 1996) where there was virtually no mathematics department and hardly any research to talk about. It was a relaxed 45-minute talk (in reality, under 40 minutes). Most of the audience were engineers (faculty and students; but the invitation came from an open-minded visiting experimental physicist).
I covered, no sweat: (0) Introduction; (I)Theorem 0 of the graph theory + Königsberg bridges Euler's theorem; (II) Euler characteristic for $\mathbb S^2;\ $ (III) Non-planarity of the Kuratowski graph $\ K_{3,3}.$
My audience was fine but nowhere as sharp or knowledgeable about mathematics as 9y old talented students.

I am willing to provide a detailed plan of the Euler characteristic portion of my talk together with the time schedule (the order and the details are important!) -- when this part is extracted and treated as a stay alone talk, it comfortably fits 15 minutes. During my lecture  I used but blackboard only. If you prepared some paraphernalia then it'd be even nicer. Do it yourself, don't let naysayers stop you.
A: You could try giving a talk in the spirit of Nets, Puzzles, and Postmen, which is a book targeted at the general audience without mathematical background, about graph theory and how it can be applied in the real world, as well as some interesting connections with deeper mathematics. For example, the book describes and justifies Sperner's lemma, and even sketches how it can be used to easily prove Brouwer's fixed-point theorem. Not many people (even those with higher mathematics background) know about that connection.
A: This summer a friend of mine gives a very nice talk about the game Nim. This is fun, you can start challenging and playing a few games with the youngests. But it also contains the deep and impressive Sprague–Grundy theorem,
A: Talk about some elementary number theory problems. Twin prime conjecture, infinitely many even perfect numbers, existence of odd perfect numbers. These are all ones which are simple enough.
A: I always found the Collatz Conjecture to be both simple and fascinating (https://en.wikipedia.org/wiki/Collatz_conjecture)
The operations are understandable to any 9 year old (who won a math competition!), and the implications are far reaching. Additionally, your college winners will understand and appreciate it as well.
It may give the younger ones motivation to think on it (the proof seems it should be sooo simple and yet it's juuuust quite out of reach... for somebody who's just starting in maths; I know I have thought about it a lot when I was young, thinking there MUST have been a simple proof for such as "simple" question).
Lastly, you can always throw in an XKCD for laughs! https://xkcd.com/710/
A: Good luck! I really like your first bullet idea.  I would also add the amazing fact that Gödel proved that we cannot prove the consistency of mathematics - the 9-year-olds will get the weirdness of this fact, especially since they must be interested in mathematics!
And then you could introduce coding.  I think they would enjoy using binary digits to code, for example, the subsets of a 3 element set (000, 001,...).  But make the original 3 element set something visual - like a pink square, a yellow triangle, and a green circle.  Find the 8 subsets, and then use 3-sequence digits of binary numbers to code the sets.
Now we have gone from the completely visual to the completely numerical!  Then, you can talk about how because of Gödel coding we have the proof of the incompleteness theorem, AND this is how computers work - by coding all of the words and everything into numbers.
Again, good luck!  I'm sure whatever you choose will be great!
