Pre-REU Research Paper, Summer 2009
You will be doing a research paper in which you pick a mathematical topic and research
it via the web, books or any other way you wish. Then you will write a paper explaining
the mathematics you have learned. The best papers will also contain something you created
yourself such as an example to demonstrate understanding of the topic. The level of the
discussion and examples should be for undergraduate students. I am happy to discuss the
project with you, including helping you with the material.
The following is a list of possible topics, courtesy of Dr. Geller. Feel free to choose
another topic as long as you check with me first to be sure the topic is suitable. Rank for
yourself the projects from most desirable to least desirable. On Wednesday, July 1 I will
question you in a random order and you will choose your topics. Each topic can be done by
at most one student, so you cannot ask for a topic if it has been selected already. The first
draft of the paper is due by 2:00 pm on Friday, July 10 at the beginning of the computer
lab and the final draft is due by 2:00 pm on Thursday, July 16 at the beginning of the
lab.
Possible Topics
1. The golden ratio
2. Fibonacci numbers
3. Polygonal numbers
4. Pascal’s Triangle and its applications
5. Rational election procedures (Can you set up a voting system that can’t be manipulated?)
6. Tilings (e.g., the chess problem of the knights)
7. Latin squares
8. Fractal patterns
9. Cryptography (secret codes)
10. Error-correcting codes, especially linear or matrix codes
11. The five color problem (All planar graphs can be colored using at most five colors so
that no two countries with a common borderline have the same color. Actually four
colors are enough but that is beyond this course.)
12. p-adic integers
13. The Koenigsburg bridge problem (or Eulerian Circuits).
(OVER)
14. Investigate and explain one of the following paradoxes:
(a) The Prisoner’s Dilemma (How to get the smallest sentence for both)
(b) The Surprise Examination Paradox (Can there be an announced surprise?)
(c) Newcomb’s Paradox (A variant of the Monte Hall problem)
(d) The Petersburg Paradox
(e) The Exchange Paradox
(f) Braess’s paradox
(g) Zeno’s Paradoxes