**Joseph Malkevitch**

*Department of Mathematics and Computing*

York College (CUNY)

Jamaica, New York 11451-0001

email: malkevitch@york.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.)

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.