FROM  RIDDLES  TO  MODERN  MATHEMATICS

MWF: 2 - 2:50pm, Steven J. Miller (Kassar 210, 3-1123), sjmiller AT math.brown.edu

Course Description:

The purpose of this course is to introduce students to many of the tools, concepts and techniques in mathematics. The starting point of many of the lectures and discussions will be various riddles. In addition to being fun, these riddles often highlight key features of the modern mathematical landscape. Also, unlike many classes where the answers are either right or wrong, often there are several answers. Moving from one answer to a better one is indicative of the refinement of theorems, and a nice respite from classes where you are either right or wrong.

We will choose problems that lead to cryptography, efficient algorithms, basic number theory and probability theory, to name just a few of the topics; other topics will be chosen based on the interests of the class. For most of the course, high school algebra suffices, though occasionally calculus will be used. While there will be standard lectures, these problems are the springboards for the lectures. As such, there will be active participation as students present various solutions.

For example, consider the following problems:

• Ten identical cookies are to be distributed among 5 different students (A, B, C, D, and E). All ten cookies are distributed. How many different ways can the five students be given cookies? HINT: you could keep track of all the different combinations: first assume A gets 10, then A gets 9, then .... but there is a faster way of doing it, that allows one to solve it in one's head, without paper, without pens! (Also, brute force is unavailable for larger starting values). This problem leads to lots of good questions about combinatorics and parition theory; for example, the famous Goldbach conjecture that every even number is the sum of two primes.

• As another example, consider the following problem: 100 mathematicians are standing in a line, wearing a black or white hat. Each mathematician can
ONLY see the color of the hats of the people in front of them. So the first person sees no hats, the last sees 99. The mathematicians are allowed to talk to each other and decide upon a strategy. Each person can only say 'black' or 'white'. For each person who correctly says what color hat they're wearing, \$1,000,000 is given to the team. How much money can the mathematicians ensure their team makes? A little thought shows you can ensure that 50% are right. Then about 66%. Then 75%.... This is indicative of the different levels problems and their solutions have.

• Finally, for an example of how high school algebra can be used for cryptography, imagine the following scenario. Consider an army with 10 generals. One wants a security system where any three of them can determine the code to launch nuclear missiles, but no two of them can. It is possible to devise such a system by using a quadratic polynomial such as a x^2 + bx + c -- to launch the missiles, one must input (a,b,c). One cannot just tell each general one of a, b or c (as then it is possible that some subset of three generals won't know a, b and c); however, if a general knows two of (a,b,c), then a set of two generals can launch the missiles! What information should be given to the generals so that any three can find (a,b,c) but no two can? What about the general situation with N generals and any M can launch (but no set of M-1) can?

For more riddles, you can check out the site I maintain at http://www.math.brown.edu/~sjmiller/riddles/riddles.html.

Class Participation:

This is a freshman seminar - class participation is essential, and will range from choosing topics to study, speaking up in class, presenting solutions or partial solutions to problems on the blackboard, and giving at least a 20 - 30 minute lecture; one of the reasons for presenting a small lecture is to learn blackboard skills, to gain experience in lecturing in public, and to learn how to give a good talk.

Books:

There following is a list of useful books for the class:

1. Mathematical Journeys (Peter D. Schumer; required): This book is a collection of accessible mathematical puzzles, and the math behind them. It is an excellent source of topics for the class, especially for the student presentations.

2. Proofs from THE Book (Martin Aigner and Gunter Ziegler; optional): The great mathematician Erdos used to joke that G-d kept a book with the best proofs of mathematical theorems; as a mathematician you do not need to believe in G-d, but you need to believe in THE Book. This is a collection of truly elegant proofs of a variety of results in mathematics. It is an excellent source of topics for the class, especially for the student presentations, and highly recommended for anyone who wants to be a mathematician.

3. An Invitation to the Theory of Numbers (G. H. Hardy and E. Wright; optional): This is one of the classical texts on number theory. We will not cover too much from this book, but it's a great source for some gems of elementary arithmetic and number theory.

4. The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary (Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil, G. Alexanderson; optional): The Putnam competition is given each year in the US and Canada. There are 12 problems; the most common score on each problem is a zero out of ten! This book is a collection of problems and solutions, as well as commentary on how these problems are related to modern mathematics.

5. An Invitation to Modern Number Theory (Steven J. Miller and Ramin Takloo-Bighash, to appear Spring 2006; optional): This book is based on 5 years of undergraduate research classes which I have taught with many colleagues. The purpose is, using as little background material as possible, to introduce the reader to accessible open problems in number theory. Parts of the book will be provided as handouts during the semester.

This course is pass-fail. There will be a take-home midterm and final, in addition to daily homework assignments and a class presentation. There will be a lot of work, but if you try hard (participate in each class, attempt each homework assignment, ...) there are enough points available so that you will pass.

Handouts:

The following handouts may be used during the semester, and greatly range in level of difficulty.

References:

The following are some useful links: