MAT 1932, Eclectic Problems in Mathematics

A Problem Solving Course at FAU

Updates How to contact me.

Goals. This course is intended to acquaint high school students with various problem solving techniques, with the expectation that the students will become more proficient at problem solving. Through problem solving, students will be introduced to mathematical topics and ideas which are not normally part of the high school curriculum, but that are at a level appropriate for talented high school students.

Students will also be expected to communicate their ideas, both written and orally. Finding a solution to a problem is an important step. Convincing others that the solution is valid is another. Elegance of the solution is also desirable, and we will frequently look for a better solution.

Instructor. Dr. S.C. Locke. Science and Engineering building, SE 286. 561-297-3350.

Class Schedule (tentative). Saturdays, 9:00 a.m. to 12:00 noon. Fall 2004.

Format. Classes will consist of a lecture by the instructor, followed by interactive (group) problem-solving by the students. At times during the class, students will be asked to present their solutions at the blackboard, to write their solutions in a mathematically correct and complete way, and to constructively critique the work of other students.

Problems will be drawn from a variety of sources including, but not limited to, national and international high school competitions, problems sections of journals, and texts.

Overview. There are several steps that one should use to solve any mathematical problem. Many of these ideas apply to problems in other sciences. The following list is based on one in Larson's book, Problem Solving through Problems:

Read definitions of any unknown terms.
Look at known results regarding these terms.
Do some examples. (Sometimes using a computer.)
Draw pictures, if applicable.
Try to find an equivalent problem.
Modify the problem. See what happens if you leave out a hypothesis, or add a new one.
Choose effective notation.
Look for symmetry or parity.
Divide into cases.
Try working from conclusion to hypotheses.
Try contradiction.
Consider extreme cases.
Simplify. Generalize. (Sometimes happens accidentally by misreading problem.)

Example: I was once working on a problem for several months, as had a couple of other people. Then, as I walked past a friend's house (he knows who he is), I thought to myself, "How would ... do it?" "He'd tell me it is trivial, so generalize." So, I thought of the generalization. That didn't help, so I simplified the new problem, to a problem slightly different from the original. I realized how trivial the new problem was. That made the generalization trivial, and that made the original problem trivial. Four months work plus a walk past ...'s house and I had a solution to the original problem that I could tell somebody in five minutes. Generalize!

Note also: Just because a problem has a nice solution, doesn't mean that somebody will find it quickly.

(To complete the story: Later that semester, the problem was placed on the final examination for an undergraduate course, with an appropriate hint.)

Find a copy of Polya's How to Solve It, and read it. It is written as if the reader is a teacher, but deserves to be read by every high school student.

For some idea of what is in the book, and what other problem solvers have added, see: http://www.math.utah.edu/~alfeld/math/polya.html, or http://www.elizabethtowncc.com/ecdweb/learnlab/polya.htm, or http://www.cis.usouthal.edu/misc/polya.html.

Prerequisites. Students should enjoy solving mathematical problems and be willing to tackle new problems and learn new techniques. The student should not be afraid to cross boundaries from one mathematical area to another in the solution of one problem. Some ability to program a computer or programmable calculator could be an asset for some problems.

Sample Problems. Knowing how eager the students will be to get started, here are some problems from a previous course, Recreational Mathematics, to keep students busy until the fall. That course was aimed at undergraduate mathematics majors, but some of the problems would be suitable for students in MAT 1932. Be careful, though: Some of them are unsolved. A few of the problems have solutions, comments or hints attached. Some, like #12, use Calculus (although Archimedes could have done it). Some are disguised theorems, with hints as to the proofs. Some you might not think of as mathematical, but more as games or puzzles.

Read. Read anything you can get your hands on that was written by Martin Gardner or Ross Honsberger. Read any of the Scientific American columns, past or present.

Short Instructor Biography. I competed in national mathematics competitions in high school and in university, placing highly several times, including being on the 1974 Putnam winning team. I continue to solve problems posed in various journals, and have also published several problems for others to solve.

Tuition. The Department of Mathematical Sciences will try to find sources of financial support for students who find it difficult to pay the tuition.

Interested sudents should contact me, or write to the Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-0991. (561-297-3340) Please include your name, school, the city and county where the school is located, and your address and telephone number.

Students should also talk to their high school teachers and counsellors before the end of this school year to see if they are eligible for dual enrollment. Dual enrollment forms need to be returned (by August 1, 2004) to the Dual Enrollment Office
Student Support Services Building (SU), Suite 201
Boca Campus, FAU

More details: see http://www.fau.edu/academic/freshman/hsdualenroll.htm or http://www.fau.edu/hsdualenroll and if you need still more information, call (561) 297-2131.

URL: http://www.math.fau.edu/Locke/courses/ProblemSolving/HScourse.htm