Here are the questions:
1. State Euler's Circuit Theorem. Explain why this theorem should be true
2. Explain Cantor's Diagonalization Argument. Give examples and explain what this argument was used for.
3. Explain the Coastline Paradox. How does it relate to the Koch Snowflake and fractals in general.
4. State Euler's Path Theorem. Explain why this theorem should be true.
5. What are the two types of infinity? Give an infinite set S, explain how to create a larger set that does not have the same cardinality. How do you know that your set has different cardinality?
6. Explain how the Fibonacci numbers can be used to approximate the golden ratio. Also explain where we see the golden ratio in nature.
7. Explain what the Knight's Tour is (open and closed). What does this have to do with Sir William Rowan Hamilton?
8a. Suppose that x, y, and z are natural numbers. Give a solution (x=? y=? z=?) if possible for x2 + y2 = z2. Give a solution (x=? y=? z=?) if possible for x3 + y3 = z3.
8b. State the Pythagorean Theorem and Fermat's Last Theorem.
9. Explain how to find the dimension of the Sierpinski triangle (make sure to include what equations you are using).