Teachers As Scholars (Steven J Miller: sjm1 AT williams.edu)
LQWURGXFWLRQ WR FUBSWRJUDSKB (Introduction to Cryptography) and Benford's Law
Monday, January 30 and Monday, February 6, 2012
Quick links to sub-pages: Cryptography Benford's Law Math Riddles
Cryptography (notes from lecture one from 2011 notes here notes from lecture two from 2011 here)
Cryptography: whiteboard photos (lecture 1) available here (zipped file available here)
Cryptography / Benford's law: whiteboard photos (lecture 2) available here (zipped file available here)
Notes from 2012: lecture notes taken by David Strasburger, RSA implementation by Tom Chiari
The ability to encode information so that only certain recipients can read it (or, conversely, to read information you are not supposed to have!) contains some of the most exciting applications of pure and applied mathematics. Since at least the time of Julius Ceasar (the title to this course is encoded with the cipher he made famous), codes and ciphers have been used to protect important information. We'll discuss various cryptosystems used over the centuries, mixing history and theory. In the course of our studies we'll discuss results from number theory, group theory, graph theory and combinatorics. This seminar is most appropriate for middle school and high school math teachers, but anyone who enjoys numbers and problem solving is welcome.
Reading list: the following files are from a book in progress, tentatively titled `The Mathematics of Encryption' by myself, Midge Cozzens and Wesley Pegden), for a general audience. Feel free to download these files, but please do not distribute further. If you want to use variants of these in your schools, just let me know. Comments on choice of topics and exposition are especially welcome; email me at sjm1 AT williams.edu. I've also posted some links to related material for those who want to read more.
Benford's Law of Digit Bias, or: Why the IRS cares about Number Theory (lecture notes from 2011 here)
How can you tell if a sequence of numbers is 'random' (and, of course, what does random mean here)? We'll discuss a variety of real world problems where the behavior of the leading digits is not what you would expect. For example, the first digit of the Fibonacci numbers or equals 1 about 30% of the time, not 10% or 11% as one might expect. The IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age -- it's a good thing there were no calculators 100
years ago! We'll discuss some of the theory and many of the applications, using this as a springboard to discuss basic probability and difference equations. This seminar is most appropriate for middle school and high school math teachers, but anyone who enjoys numbers and problem solving is welcome.
I also maintain a math riddles page. Please feel free to share these riddles with your colleagues and your students, and let me know if there is anything I can do to make the site more useful for you and your classes. The goal is to add a student / teacher's corner sometime in the spring to facilitate using these in classrooms. If you're interested in helping, or want updates on the progress, let me know.
Lecture Notes from 2011
Here are lecture notes from the 2011 version of the course. These will be modified a bit for 2012, but give a sense of what will be discussed.