This page is meant to provide a collection of project ideas for undergraduates who are interested in exploring mathematics outside of the classroom.
I have worked on this project previously with Victoria Jakicic. In the work with Victoria, we were able to determine for what polygons a magic polygon exists, as well as to determine several properties of these magic polygons (namely the magic sum as well as the center number). To check out the paper, please visit the ArXiV.
Graceful Tree Conjecture
A graph is a collection of vertices and edges connecting these vertices, and a tree is a special kind of graph that doesn't have any cycles or multi-edges. In this conjecture, it is hypothesized that you can label the vertices of a tree with the integers 0 up to m in such a way that the labeling of the edges via assignment of the absolute value of the difference of the labels of the vertices causes the edges to be labeled by the integers 1 up to m. For more information, check out this website.
Magic Square of Squares
A magic square is an array of numbers have n rows and n columns such that each row, each column, and both of the diagonals sum to the same number (the magic sum). Does there exist a magic square with 3 rows and 3 columns that is composed of distinct perfect squares? For more information on this problem, click here.
Graph Theory and Board Games
The board game Ticket to Ride has a game board that is a graph, i.e. a collection of vertices (dots) and edges (line segments connecting these dots). There are many strategic game questions that can be analyzed using graph theoretic questions. This would game would create many opportunities for undergraduate research projects . . . and is a project that I am VERY interested in!!!
Earth-Moon Coloring Problem
The Four Color Theorem is a theorem that says that for any map, four colors is enough to color the map in such a way that no two neighboring areas are colored with the same color. This brings about the following question: What is the maximum number of colors needed to color countries such that no two neighboring countries have the same color in the case where each country consists of one region on earth and one region on the moon? For more information on this problem, click here.
This one involves some group theory, so it would be a good idea to have taken Abstract Algebra before working on this one (and it just so happens that I am teaching Math476 this semester . . . hint hint). Let G be an abelian group of odd order, and let A and B be subgroups of G having the same number of elements. Then the elements of A and B may be ordered in such a way that the sums of pairs of the form a+b (where a is an element of A and b is an element of B) are pairwise distinct. For more information on this problem (and a more precise statement of the problem), click here.