TITLE: Short Course Suggestions For High School Students
QUESTION [13 upvotes]: I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are familiar with single variable calculus and very basic linear algebra over the reals. The teaching format will be two hours per day, six days per week and two weeks in total. Suggestions for one week courses are also welcome.
There are two things I want about this course. First, it should have a direction and a final goal. So it shouldn't be based on  isolated Olympiad-type problems. Second, it should introduce at least one new concept or object which is not a part of the high school curriculum.
For instance, classification of frieze patterns is a good topic. There is a clear goal and one needs to introduce the concept of a group which is new for high school students. Any other suggestions?

REPLY [3 votes]: You can do Monsky's theorem, that a square cannot be divided into an odd number of equal area triangles.
On the way you will have to do

p-adic numbers, 
Sperner's lemma,
present $\mathbb{R}$ as a vector space over $\mathbb{Q}$.

In case if you have more time, you could 

use Sperner's lemma to prove Brouwer's fixed point theorem.
use (3) to do Dehn Invariants 
and I am sure you can find what to do with p-adic numbers