PREREQUISITE: Not taking or having taken Math 247, 317, or 353. Since it is more fun for students to discover fundamental mathematical principles for themselves, this course is not recommended for students who have taken or are taking these three courses. Indeed, Math 198 may serve as good preparation for more advanced courses by introducing the student to a certain way of thinking independently about mathematics.

The course covers a broad range of topics from classical mathematics with an eye toward learning how to explore mathematical reality with an open and alert mind. We will learn how to conduct mathematical experiments (perhaps aided by computers), discover mathematical facts, and construct mathematical proofs. There will be no lectures or reading material; instead, students will work together in small groups on assigned problems. Successful groups will explain their solutions to the class, and each student will prepare independently a written exposition of the problem and one of the solutions presented to the class.

Some of the problems we tackle are little more than brainteasers or puzzles. Other problems are deeper, with a wider relevance to the world of modern mathematics. Here are a few examples of topics we will consider:

Consider the following game. There are two players and some piles of beans on the table between them. A turn consists of removing some beans (at least one) from one of the piles. The loser is the one who can't make a move because all the beans are gone. Play the game with your partner. Develop a strategy. Do you find that there are some sets of piles which don't affect the outcome of the game? Which ones? Develop a general strategy for the game and prove that it works.

It has been known since the time of Euclid that there is a regular polyhedron called the icosahedron with twenty triangular sides. Most students have only a vague idea about what a regular polyhedron is. By thinking about the icosahedron, we will invent for ourselves a suitable definition of regular polyhedron, decide whether such polyhedra exist, and discover ways to construct them.

Given a regular polygon with 400 sides, show that it can be tiled with parallelograms. Prove that, if you are presented with any such tiling, at least 100 of the parallelograms are rectangles.

Consider an island in the middle of the ocean, and assume that every point on the island is on a hillside, is an isolated mountain peak, is an isolated valley bottom, or is a simple saddle point. There are no lakes on the island. Draw maps of several such islands, and try to find some correlation between the numbers of geographical features of each type that can occur.

There is a popular encryption scheme called RSA which is implemented by older versions of a program called PGP. Using the web, find out how this scheme works and explain it in class. Why is it called a "public key" system? If you like, obtain the program, register your public keys, and send each other encrypted messages. What is a digital signature?

Other topics that might be included, according to student interest, include: various other games, probability, notions of number and area, numerical topological invariants, infinite numbers, prime numbers, the mathematical basis of public key cryptography, irrational numbers, factorization of integers, the mathematical basis of algorithms for correcting errors in the transmission of digital data over noisy channels, factorization of polynomials. The successful student will learn to think about mathematics often, independently, and pleasurably.

Instructor: Daniel Grayson received his Ph.D. in mathematics from the Massachusetts Institute of Technology in 1976, then taught at Columbia University for five years. Following a year at Princeton's Institute for Advanced Study, he joined the U. of I. faculty in 1982. At Illinois, he has been a University Scholar and has also had an appointment at the Center for Advanced Study. Professor Grayson's special area of interest is algebraic K-theory (basically the study of very large matrices of polynomials via topological methods), and he is also interested in algebraic geometry and number theory. He is one of the original authors of Mathematica, and is has developed a software system called Macaulay 2 aimed at research in algebraic geometry.