PREREQUISITES: solid grasp
of calculus and working knowledge of basic
analysis and algebra, but most importantly,
a willingness to think about challenging math problems.
TEXTBOOK: Proofs from the Book, 3rd Edition, by
M. Aigner, G. M. Ziegler (Springer, 2003). This delightful book will
serve as the source for some of the more advanced
topics. I will distribute my own notes
for more introductory material.
There are many good problem solving books on the market, here are four of them:
The Art and Craft of Problem Solving
(2nd Edition) by Paul Zeitz (Wiley, 2006),
Putnam and Beyond
by Razvan Gelca and Titu Andreescu (Springer, 2007),
Problem-Solving Strategies by Arthur Engel
(Springer Verlag, 1999) and
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities
by J. Michael Steele (Cambridge University Press, 2004).
Why would you take this course?
Here are four possible reasons:
Solving mathematical problems is fun and keeps you sharp. You can
also participate in mathematical
contests.
You can learn some rarely taught topics, as well as
how to organize and present a mathematical argument.
Say you want to apply for a job at a high tech company (no
names will be mentioned on this page - I provide no
free plugs). It has become very popular
in the last 10 years or so for the interviewers in such
companies to test job applicants with puzzles. Some of these are technical
questions and pertain to the job description. (Example for programmers:
How would you find a loop in a linked list of arbitrary length
using a constant amount of memory?) Other questions are
impossible and presumably ask for an order of
magnitude estimate. (Example: How many manhole covers are there in the USA?)
But the majority of puzzles are mathematical in nature and
many will be featured in this course. Here are a couple of sources
I'll be using:
Sells Brothers,
Techinterview.
If you think you may teach one day,
keep in mind that puzzles and problems are great instructional tools.
GRADE will be based on the following:
About once a week, a few problems will be assigned as homework
due at the next class meeting.
You can work on these problems alone or in groups. You have to write
out your work alone
and turn it in. Everything you turn in has to be typed
(although you may draw diagrams and figures by hand), preferably in
some version of TeX. Some of the lecture time will
be devoted to solutions presented to the class
by the students. You have to volunteer for a few
of these presentations
during the quarter. (I may call
somebody up if there are no volunteers.)
There will be no penalty for
incomplete or incorrect solutions. Mistakes
and misconceptions are integral part of problem--solving.
Check here for a couple of sample problems,
with solutions.
You will be expected to give a lecture (about 20 minutes) on the
material in Aigner-Ziegler book.
By 3pm on Monday, March 17 (last lecture),
you have to write complete (and correct)
solutions to five problems labeled by (*) in the assignments.
These problems will not be discussed in class. You may also write solutions to any
other assigned problem not discussed in class. Choose solutions you are
most proud of!
Class attendance is mandatory and will be recorded at
the beginning of each class. For every lecture
you miss you have to turn in an extra solution to a problem labeled by (*).
However, you should
not expect pressure to excel at problem solving. This is
supposed to be a fun class, devoted to little nuggets
that are rarely covered in standard math classes. If you
are excited about mathematics, you will be able to
contribute to the class, even if you do not solve many
problems on your own!
For students interested in working with high school students who train for math
competitions, please contact ARML by email (arml@math.ucdavis.edu) or visit
their web site for more
information.
Some web sources:
Putnam comptetition
archive.
Also check this site,
maintained by John Scholes.
Some nice puzzle sites: Cut the Knot,
Nick's Mathematical puzzles,
MathPuzzle.
The International Math Olympiad page.
Some links to American high school math competitions:
USAMO archive,
American Invitational Mathematics Examination (AIME),
Bay Area Mathematical Olympiad.
Some nice problem-solving resources from other universities
(these links come and go, so be prepared for some dead ones):
Harvey Mudd College,
Stanford University,
Dalhousie University ,
MIT.
Also check out
Art of Problem Solving.
TeX software and information:
MikTeX (TeX system
for Windows), WinEdt
(TeX Editor for Windows), TeX Users
Group (TUG) (information for all level of TeX users),
GSView and
Ghostscript.
A LaTeX textbook.