**Joseph Malkevitch**

*Department of Mathematics and Computing*

York College (CUNY)

Jamaica, New York 11451-0001

email: joeyc@cunyvm.cuny.edu

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.

**Project 1: Triangulated Polygons**

**Project 2. Nets for Deltahedra**

**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 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 14: Drawings of 3-Polytopes**

**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**

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

**Acknowledgements
**

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.