Interesting piece of math for high school students? I'm giving an hour long lecture to high school math students with a fairly high aptitude in math. I want to present something a little advanced for them (undergrad level) that they have to struggle with. So I want to be able to present a fact and prove it, or present an interesting theory in a single standalone hour long lecture. Any ideas as to what I should speak on?
 A: I suggest graph theory: an one hour lecture can for example start with the bridges on Koningsberg and move up to proving that a graph contains an Eulerian cycle if and only if every vertex has even degree.
A: Try presenting Goldbach's conjecture, and let them have a go at it. :)
In all seriousness, they should get an introduction to these types of open problems, and your class may be it. An apparently easy problem can actually turn out extremely difficult, as is evident in such a conjecture. Others could include the Collatz conjecture, which is also interesting, or the Riemann hypothesis (which needs a little more mathematical understanding). 
A: A few suggestions:
$1.$ Hyperbolic Geometry. This gives you the opportunity to explain what an axiomatic system is (something rarely discussed in high school geometry), and then explain some of the interesting results that arise when Euclid's fifth postulate is false. For example, it is very simple to prove that Euclid's fifth postulate is equivalent to the sum of degrees in any triangle being $\pi$. So what can the students say about the angle sum of triangles in hyperbolic geometry?
You can mention models of hyperbolic geometry and show the Circle Limit pieces of M.C. Escher, this is very tangible and the pieces are beautiful. Mention that they are tilings of the hyperbolic plane with regular polygons (yes, the "smaller distorted polygons" at the edges are all congruent to the one in the center).
$2.$ Fractals. It is easy to talk about measuring the length of the coast and start talking about fractal dimension. You can talk about simple fractals like the Sierpinski triangle. Using some simple series you can calculate the area and the perimeter of the fractal after $n$ iterations. Take the simple limit as $n\rightarrow\infty$ and you can see some of the bizarre properties of fractals. Also, the Mandelbrot set has some fascinating properties, check out Farey addition.
$3.$ The game Hex. This game is fun! Also, the game can be used to complete a proof of Brouwer's Fixed Point Theorem in two dimensions. The proof uses only the rules of the game and is a nice way to provide into higher math.
$4.$ Taylor Series. If the student are advanced, they may have had some calculus. You can use Taylor series to prove Euler's Identity. This is the first piece of mathematics I found beautiful. It had a big impact on me. Also, Euler's solution to the Basel problem is simple with Taylor series.
$5.$ Infinity. There are tons of routes you can go with this.
A: Another (possibly controversial) option would be to introduce them to the Pi vs Tau debate (perhaps split the audience to take sides). It could be a fun way to explore a broad range of mathematics.
http://www.tauday.com/
A: Compute Pi in creative ways.
If it's raining, via the Montecarlo method. Draw a circle on a cardboard, place it in the rain for a few seconds and then count the total number of drops and the portion that actually got inside the circle. Your lecture could go on explaining Montecarlo simulations. Some ideas can be found here: http://en.wikipedia.org/wiki/Monte_Carlo_method
If it's not raining, toss a handful of pins in the air and then use Buffon's method to compute pi. http://en.wikipedia.org/wiki/Buffon%27s_needle
If your students never heard about Buffon's needle, they will surely look puzzled when you perform that act as first thing after entering the classroom.
A: I'd suggest giving a lecture on How to prove things. Generally this is not properly covered in high school (at least in the UK equivalent), and it's not until you get to university that you discover that you don't really know what proof means.
It would make a nice lecture to take them through:


*

*Proof that $\sqrt{2}$ is irrational.

*Proof that there are infinitely many primes.

*Proof by induction.

*Proof of various basic set-theoretic notions.


The last one is where we started at Oxford for undergraduate material. We were given a few things to prove (that $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$, etc.). We all went off and did our homework and came back with some pretty Venn diagrams, and got laughed at, and shown why that was wrong and how to do it properly. It was quite eye-opening and great fun.
A: I gave such a talk once and focussed at the applied end of mathematics, introducing them to polar coordinates and elementary ODEs. 
In 2-D and 3-D polar coordinates the mystery as to why the surface area $4\pi r^2$ is the derivative of volume $\frac{4}{3}\pi r^3$ and circumference $2\pi r$ of area $\pi r^2$ is solved. This is something that immediately set off some light bulbs and demonstrates some elegant simplicity compared to the Cartesian coordinate derivations they had seen to date.
Then this puzzle which they enjoyed: 

Imagine a fighter jet traveling at Mach $m > 1$. What should its path be so that the shock wave hits a target all at the same time? 

Set up the ODE and solve to obtain a logarithmic spiral. Then show examples of logarithmic spirals in nature.

Whatever you decide to do, good luck! It's great you're doing this.
A: Two actually useful topics you could talk about are:


*

*Calculus of variations (if they're more mathy). Prove that the shortest path between two points is a straight line. Then generalize it to an arbitrary index of refraction. Alternatively/additionally, introduce them to the brachistochrone problem and solve it for them.

*Error-correcting codes (if they're interested in CS too). I did this with a class containing a mix of middle- and high-school students, and at least some of them were able to follow along and ask great questions.
A: When I was at high school, whenever I asked "why do i need to know this" the teacher would respond "for the test". I wish they could have show me solid applications. So I would focus on things that you can show applications of that they can relate to. 
A really easy one would be relating it back to computer games. 
As a computer graphics programmer here are some of the things that I wish I had had a solid foundation when I left high school:  


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*How to find out if 2 objects are colliding (from simple spherical collision to complex polygons to the separating axis theorem

*Matrix/Vector translations, rotations etc

*Quaternions and how they solve Gimble lock

*A Star algorithm and other navigation graph solutions 


I don't know the level of your students and looking at the other answers, my suggestions may be far too simplistic and less applicable to students that already have a taste for mathematics.
I provided some links with nice interactive examples that might be good for presentations. But going one further and showing their use in games might be good. The first link are tutorials from the makers of the hugely successful NGame so that might be fun to show them (thewayoftheninja . org)
A: As a high school student, I would say that I would find the following the most appealing:


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*Ramsey Theory: You can start with the "Happy Ending Problem" and then introduce the idea of different pigeon hole principles and end with VDW theorem for some simple cases.

*Surreal Numbers: Even though they are practically useless, they are neat when you involve infinities, ordinal, and cardinal numbers.

*Mandelbrot and Julia Sets: One could waste hours zooming in to the Mandelbrot set alone. It would be neat if you explained how they worked, and introduced generalizations such as using a cubic iterative function instead of a quadratic etc.

*Cantor Set: This topic alone could be an entire class by itself. You can introduce a bit of topology and measure theory to give an idea of why the cantor set is so cool. You could also explain some extensions of it like the Devil's Staircase (a function with derivative 0 almost everywhere) or other weird functions such as Conway's Base $13$ function. 

*Calculus of Variations: If your audience is familiar with calculus and some physics, you could introduce the Brachistochrone problem as well as proving why the shortest path between two points is a line. 

A: Thom codes are a very simple and nice application of elemental differential calculus. See Manipulation of real roots of polynomials: Isolating Intervals or Thom's Codes.
A: I too suggest graph theory, but at the level of proving Mantel's theorem on the maximal number of edges in a triangle-free graph, or Prim's algorithm to find the minimum spanning tree of a weighted graph.  If these students are as adept as you say, both these topics would be suitable for a one-hour undergraduate level lecture.
Indeed, many problems in combinatorial analysis (of which graph theory is but one part) are amenable to discussion at a relatively elementary level.  Latin squares, block designs, generating functions, and error-correcting codes all contain simple theorems and ideas that can be discussed without prerequisites in analysis or abstract algebra.
But, high school students being who they are, I would imagine there are very few such students who would appreciate an hour-long theorem-proof lecture of the sort you might have in mind, irrespective of their level of mathematics proficiency.  What would get more attention is less focus on proof and more focus on interactivity, results, and novelty:  if the result is counterintuitive, for instance, that would tend to generate more interest.  Even more so would be the discussion of open questions that have elementary statements--e.g., the Collatz conjecture.
A: I had a very similar experience when I was in lower 6th form in England (16-17). So this is just my experience and I thought was useful to mention (hopefully useful).
I did a session on $\textbf{nonlinearity}$ and $\textbf{chaos theory} (at the time it was crazy but pushed me hard to understand). The premise was to try and understand the different patterns that arise in animal coats for example was an exciting part. If I also remember correctly, we looked at the compound pendulum and the breaking of a beam. 
So maybe something along those lines. 
To wrap up the story, I eventually went onto study the topics I mentioned at undergrad and then phd, where I used those same notes (but obviously a bit more maths ;) ). 
Good luck 
A: I guess the first question to ask would be what are you trying to achieve or demonstrate. Possible answers:


*

*maths are (can be) fun $\implies$ pigeonhole/Ramsey theory, paradoxes, fractals ... (I realize that "fun" is very relative)

*maths (beyond the abacus) are actually useful in real life $\implies$ Cauchy-Schwarz implies Heisenberg principle in quantic physics (maybe a bit advanced), Ito theory implies Black-Scholes (implies money) in finance ...

*maths are tough (I'm not sure why you'd go for that one)

*maths are an active/open field $\implies$ open problems: Goldbach conjecture ... or recently solved problems: Fermat LT, 4-color theorem ...


Now for totally subjective examples of what I may have enjoyed seeing when I was in high school:


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*Brouwer's fixed point and hairy ball theorems: 


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*the context is easy to grasp (for the FPT, you may actually refer to the story of Brouwer getting the intuition while watching coffee swirl in a cup)

*you can leave the result as a question for a minute or two, but it is easy to get an intuition of the result with some mental play

*and then you can show that (1) proving it is far from trivial and (2) it can be proven using widely different approaches (you can look at this paper)

*obviously you can only give sketches of proofs but I doubt the public would enjoy full-blown proofs of anything anyway


*Any subject related to computers or information theory (betting that your "students with a fairly high aptitude in math" may be on the nerdish side):


*

*JPEG compression: you can use it to mention Fourier transform/DCT (and lossy compression) and Hoffman encoding (lossless compression). You can even open up on better formats/algorithms such as JPG2000/Wavelets

*Shannon's theorems on transmission rates (I miss good examples here)

*Cryptography and how it relates to number theory


