Nature of Student Activities:
The proposed project differs from typical REU Site proposals in the fact that all the
participants will be prospective or practicing teachers. No attempt will be made to “recruit” the
participants into graduate programs and turn them into mathematicians. The main goal of the
project is to enable the participants to experience firsthand the thrill of creating/discovering
mathematics. The intent is to make it a lasting experience that they can draw back on, be proud
of, and easily relay to their students. The greatest impact of the project will be realized if the
participants are able to replicate the experience in their own classrooms.
With that in mind, some of the explored topics will be selected as to have “transfer benefits”
to the participants’ classrooms. Ideal topics would be relatively simple to explain, would have a
rich history, “big” conjectures and a large amount of known results and remaining possibilities. A
significant part (upwards of 30%) of the project will be taken up with the preparation of quality
teaching modules on these topics. The rest of the time will be spent in the pursuit of original
results. While it is not the primary goal, nor the primary measure of success of the proposed
project, it is likely (based on the expertise and experience of the PI) that original publishable
results will be obtained by some of the participating teams. The explored topics are chosen to
increase such likelihood.
Some of the possible topics are described below.
 Scheduling Social Golf Outings. This is an example of an immersion problem; a problem
that can be easily stated, in layman terms, and can be easily varied. Suppose 12 friends
gather annually to play golf in groups of four. How many rounds should they play to
ensure that each person plays at least one round with each other person? Now suppose
they can only play a limited number of rounds. How should they be grouped in order to
maximize the number of distinct pairings (i.e., maximize the number of interactions on
the course)? Is the maximal solution equitable? Equitable here means that Person A has
basically the same schedule as Person B. How does this problem generalize? What
mathematical tools are needed to solve it? In what cases can equitable maximal solutions
be attained?
 Generalized Oberwolfach Problem. Another immersion scheduling of social events
problem. Suppose 15 people are attending a conference. They will eat seven meals
together. The dining area has two round tables: one seats eight, the other seats seven. Is it
possible to arrange the seating so that each person sits next to each other person exactly
once? How do you generalize this problem? What if the number of participants is even?
What if four of the meals are held around a table for 15 and the other three meals are in
the original room? Now how does the problem generalize? What is the corresponding
graph theoretical concept? This problem is appropriate for a teaching unit.
 Latin Squares and their Applications. This is a broad area with an extremely rich and
fascinating history and a large number of practical applications. Latin squares have
applications in experimental designs, affine and projective designs, tournament designs,
secret sharing schemes, graph factorizations, and many other areas. The study of Latin
squares provides a great opportunity to learn about conjectures, proofs and
counterexamples. This is an ideal area for developing a teaching unit.
 Graph Labelings and Cyclic Graph Decompositions. The PI is a leading expert on this
popular topic and is responsible for proposing several large conjectures in this area. Alex
Rosa introduced Labelings in 1965 as means for attacking the conjecture that every tree T
with n edges decomposes the complete graph K
(known as Ringel’s Conjecture).
2n 1
Most of the publishable results obtained by the PI’s students were on this topic.
Programming opportunities abound in this area. This is another appropriate area for the
development of a teaching unit.

Other possible topics include: Public Key Cryptography (appropriate for a teaching module),
Affine and Projective Geometries (also appropriate for a teaching module), Balanced Incomplete
Block Designs, and Tree Decomposition of Regular Graphs.
All the above topics will allow the PI to expose the participants to very interesting and
important concepts. Moreover, the topics have the property that they can be introduced without an
excessive amount of background material. This should allow the research teams a degree of
independence as the project progresses. Moreover, the participants will learn to utilize
computational techniques to gain further insight into their research problems. Computer software
packages (such as Maple and Visual C++) will be used as exploratory tools for the projects and
typesetting packages (such as Eclipse) will be used to prepare student reports. The participants
will have access to the computer labs in the Department of Mathematics at Illinois State
University, which are equipped with the necessary software.