Trying to cover the material in a course is usually bad. You don't have much time, and even if the students learn the material, they might just be bored when they see the material later in a normal class.
I think you have an opportunity to let the students do some mathematics, and to see some of its beauty and challenge. Don't make the mistake of trying to get them to work on open problems, but do give them activities related to the following parts of mathematics:
- Guess what is true.
- Prove what you believe is true.
- Communicate.
It was a big surprise to me when I learned in the PROMYS program that these are what mathematicians do as opposed to the types of things I had seen in mathematics classes, which concentrated on learning techniques and applying them. I do not recommend trying to follow PROMYS because you have much less time.
For example, you can start with Pascal's triangle. Have them look for patterns. Suggest looking at the even-odd pattern, or the sums of every element in a row, every other element or every third. Estimate the size of $2n \choose n$, and ask how you might compute ${30 \choose 15} \approx 2^{27}$ with $32$-bit numbers so that you can't compute $15! \approx 2^{40}$ directly. (Recursion? ${30 \choose i}/{30 \choose i-1}$? Prime factorization?) Ask for generalizations to multinomial coefficients. Show some connections to other areas of mathematics, e.g., ask how many faces a hypercube has of each dimension, or mention the Central Limit Theorem. You can show how you might prove some of these patterns with induction, or bijective arguments, or by evaluating $(x+y)^n$ at particular values of $x$ and $y$. Have them read an article covering related material in something like the College Journal of Mathematics or Quantum so that they see some good exposition, and that mathematics is still active. Let them flesh out and present parts of the article. Then have them write something about what they have learned.
An advantage over some competitions which emphasize distinguishing the most exceptional individuals is that there is something for everyone in Pascal's Triangle (or in the exploration of similar rich subjects of mathematical study). If your students have a lot of talent and energy, they can work on complicated material. If they are not as advanced, then they can still make real progress on simpler patterns.
An historical overview in parallel might complement the hands-on study.
Extending on Ralph's answer, there is a similar very neat proof for the formula for $Q_n:=1^2+2^2+\dots+n^2$. Write down numbers in an equilateral triangle as follows:
1
2 2
3 3 3
4 4 4 4
Now, clearly the sum of the numbers in the triangle is $Q_n$. On the other hand, if you superimpose three such triangles rotated by $120^\circ$ each, then the sum of the numbers in each position equals $2n+1$. Therefore, you can double-count $3Q_n=\frac{n(n+1)}{2}(2n+1)$. $\square$
(I first heard this proof from János Pataki).
How to prove formally that all positions sum to $2n+1$? Easy induction ("moving down-left or down-right from the topmost number does not alter the sum, since one of the three summand increases and one decreases"). This is a discrete analogue of the Euclidean geometry theorem "given a point $P$ in an equilateral triangle $ABC$, the sum of its three distances from the sides is constant" (proof: sum the areas of $APB,BPC,CPA$), which you can mention as well.
How to generalize to sum of cubes? Same trick on a tetrahedron. EDIT: there's some way to generalize it to higher dimensions, but unfortunately it's more complicated than this. See the comments below.
If you wish to tell them something about "what is the fourth dimension (for a mathematician)", this is an excellent start.
Best Answer
I made these ropes with rare earth magnets in the ends for demonstrating knots. The materials (rope, magnets, PVC pipe and glue) are inexpensive. I've used Tangle before to play with knots, but it doesn't tend to move over itself very easily and it can be hard to see at a distance which strand is on top at a crossing.