Q & A with Dr. Reiter
What math class did you take as a 9th grader?
I took Geometry in ninth grade, just like everyone else. It was a good class at a good school (in Lafayette, Louisiana) and I learned what it means to prove a theorem. Today we don't teach the course in this way. We didn't have acceleration, but the courses were what today would be called enriched.
What is the best way to study for a math competition in general? For MathCounts?
Do problems. Put yourself in the same position as a competition participant and take an old contest using the same time allowed during the contest. Then learn about the problems you've missed.
|
|
Harold Reiter, PhD
Hometown: Shreveport, Louisiana
Education:
PhD, Mathematics, Clemson University
Interests: travel, spending time with my one- and three-year-old grandsons
Free-time Favorites: reading light novels (like John Grisham), running, racquet sports, watching or playing bridge at Bridge Base Online
Favorite treat: Dark Chocolate
|
What is your favorite "principle" in mathematical problem solving? Is it the invariant principle?
My favorite idea for problem solving is to 'trade in' a hard problem for an easier problem. Then make the easier one a little harder as I get a better understanding.
What are your favorite math questions?
I have two favorite MATHCOUNTS questions, both from MATHCOUNTS Handbooks. The first is one about the difference between number and numeral. Here it is.
Consider the following list.
Are these numbers getting larger or smaller?
A few years later, I was asked to write a handbook Stretch about
combinatorics. What appeared was a sequence of 10 questions each of
whose answers was the following binomial coefficient. 
I picked six different ways to model the problem, but the solution was the
same.
As for higher level problems, check out my column in Mathematics
and Informatics Quarterly
and my problems at the London
Sunday Times.
What is the most interesting MATHCOUNTS question you've written?
Although it has been a few years since I've written MATHCOUNTS problems, one that comes to mind is a shortest path problem on a many sided polyhedron. The net of faces is given and the endpoints of the path are given. I think some students actually built the polyhedron to solve the problem.
What thought processes do you use when creating particularly
enjoyable math problems?
I like to try creating problems in bunches of 8 to 12 using a
combination of two or three ideas. I'm particularly fond of counting and
existence problems that use (a) the inclusions/exclusion principle, (b)
polygons in the plane all of whose vertices are integer lattice points,
and (c) the Pythagorean Identity. Here's an example. How many squares in
the plane have two or more vertices in the set {(0,0), (0,1), (1,0),
(1,1)}?
What kind of topics and thinking do you try to cover with your
questions?
I enjoy problems that combine two or three ideas. I play around with
number theory problems that involve number of divisors, sum of digits,
and product of digits. I'm especially fond of problems that hinge on a
repeated process that has an easily discovered invariant. See the M&IQ
problems mentioned in my first answer.
How do you think of all of the interesting properties of certain groups of
numbers that make the problems work? Do you start with a group of numbers
and look at properties, or do you think of an interesting property and try to
find numbers that work with it?
Usually, the latter. For example, suppose I'm thinking about problems involving remainders. I would like to test students' understanding that you can compute the remainder of a sum by first taking the sum of the remainders. Now what set of numbers could we use for such a problem? Of course, Fibonacci (or Lucas) numbers! The problem I write is: What is the units digit of the 2007th Fibonacci number?
How is mathematical problem solving related to mathematical research, and
what is the relationship between the two? How does one combine these
sometimes seemingly different creative endeavors in a harmonious and fruitful way?
These are hard questions. For some mathematicians, problem solving IS
research. For me, that is sometimes the case. For others, problem
solving might come into play, but the focus might be on theory building.
In that case the problem solving is peripheral. The excitement and
satisfaction we all get from reaching inside ourselves to find ideas
that we didn't know were there is certainly part of both research
mathematics and creative problem solving.
More about Harold Reiter…
Reiter's daughter Ashley drew him into the world of composing when, as an inquisitive sixth grader, she'd ask him to make up problems for her as they were driving in the car. "We began a family problem solving adventure," says Reiter. Reiter was composing puzzles for Ashley regularly, even on family vacations. "She liked it, so I began to
like it too. I realized that I could write good problems and it was fun." 
Ashley was the first girl to make it to the top 10 in the MATHCOUNTS national competition (1987), and she grew up to earn a PhD in mathematics herself (See the article The Unity of Mathematics that she wrote for Imagine magazine in 1998). Hooked on composing, Reiter began writing problems for the American Mathematics Competitions, then volunteered to write for MATHCOUNTS in 1990, starting his first four-year term on the seven-member committee in 1991.
Although Reiter no longer composes for MATHCOUNTS, he is still passionate about the program and problem solving. When I caught up with him, he was just back from a jam-packed weekend of coaching eight North and South Carolina middle-schoolers in preparation for the MATHCOUNTS national finals on May 11, 2007. Although weekends like these admittedly wear out the soon-to-be 65-year-old Reiter,
he has no intention of stopping. "I have no plans to retire," he says, "my work has never been more satisfying and enjoyable."
Related Forums
Discuss Math, or MATHCOUNTS, share some Math Humor, or solve some problems.