Mathematics Research Projects

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451-0001

The projects below are designed for use by middle school, high school, and undergraduate students. The purpose of the projects is to call the attention of students who might have an interest in working on their own to mathematics projects in areas that have not been as "heavily mined" as some other topics that have served for such investigations in the past. In many cases the answers to some (or all) of the questions posed as part of these projects are already known. However, it is common for students who work on projects of this type to see new issues that may not have been answered in the past. This is how mathematics grows and evolves. In any case, the purpose of posing these projects is to spur additional interest on the part of students in trying to work on their own on problems that are at the fringe of the traditional curriculum, thereby expanding their mathematical horizons, and increasing their interest in and appreciation of mathematics.

The sources of these problems are very varied, including some that I believe I have originated. I have not given explicit credit for the origins of each problem so as to encourage students to start thinking about the problems on their own rather than first looking at what is known about the project. For each project I have added at least one problem that to the best of my knowledge has not been posed before. I would like to thank those individuals who created some of these questions for their inspiring work.

I welcome additional projects to supplement those shown here as well as feedback about the experiences that students (or the teachers who supervised them) might have while attempting to work on them. Some of the terms used in the project descriptions which may be unfamiliar to students are defined in a glossary at the start of the project descriptions.

Please feel free to copy and distribute these problems provided the full text shown for each problem is retained.

Projects Glossary

Project 1: Triangulated Polygons

Project 2. Nets for Deltahedra

Project 3: Orienting Graphs

Project 4: Hamiltonian Circuits Arising from Coded Paths in 3-valent Plane Graphs

Project 5: Coloring Platonic Solids

Project 6: Constructing Polyhedra with Tabbed Panels

Project 7: Disjoint Triangles

Project 8: Vote Matrix (Table)

Project 9: The Geometry of the Core of a 3-Person Game

Project 10: Triangulation of Plane Polygons

Project 11: Visibility of Equilateral Polygons

Project 12: Cuboids

Project 13: Drawing Graphs

Project 14: Drawings of 3-Polytopes

Project 15: Nets of Bricks

Project 16: Repeating Decimals

Project 17: Nets and Truncation

Project 18: Spanning Tree Versus Graph Distance

Project 19: The Center of a Plane Graph

Project 20: Polyhedra Folded from Regular Polygons

Project 21 : Polyhedra Folded from Isosceles Right Triangles

Project 22: Permutation Primes and Cyclic Primes

Project 23: Carousel Numbers

(Comments and results related to these projects are welcome, as well as suggestions for additional projects.)


Some of this work was prepared with partial support from the National Science Foundation (Grant Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning (administered by SUNY at Stony Brook, Alan Tucker, Project Director).

Comments and improvements from Branko Grünbaum are gratefully appreciated.