#### Foreword

With all the push toward applications of mathematics where some are at best artificial, it is refreshing to find a text that does not have the pretense of giving any real applications, but rather a book on number theory just for fun. The conception of the book, *Numberama*, could have been conceived by the first real number theorist, P. Fermat, through a bunch of problems (without any thought of applications).

This book is a text about problems in number theory intended to aid teachers from early elementary school to early high school in giving an appreciation of number theory to their students. The text is divided into four parts and an Appendix: Chapter I is devoted to several basic problems in number theory which can be appreciated using only elementary arithmetic: addition, subtraction, multiplication, and division. Chapters 2, 3, and 4 are devoted to board games based on the problems in Chapter 1, where each chapter requires successively higher level arithmetic skills to play the games in the chapter. For each of the problems given in Chapter I, there is a code indicating the level of skills needed to work on at least parts of the problems. Teachers are given plenty of advice as to how to present the material to the students. The Appendix includes excerpts from various student and teacher participants in Dr. Benjamin's Numberama program, which describes both the benefits and the joy they received from participating in this program.

A couple of the problems included are the following, (some going back to antiquity): Finding perfect numbers. Here the students get to use their skills at multiplication and division, as well as being exposed to prime numbers. This problem leads, of course, to open questions such as the existence of infinitely many even perfect numbers and also the existence of at least one odd perfect number. There is also the problem of representing integers as the sum of two squares, a problem which Fermat himself worked on. Again the teachers are given hints as to how to proceed.

Chapters 2, 3, and 4 are devoted to 19 different board games concerning the problems in Chapter I, which seek to hone the skills of the students, involving many of the properties of numbers given in the first chapter. The text is written well and seems to be accurate. I would certainly recommend it to teachers interested in enriching the mathematics content and honing basic arithmetic skills of the students.

**Chip Snyder**

Professor of Mathematics,

University of Maine,

Orono,

Maine

#### Preface

It has been nearly 30 years since I began working on my *Numberama* book. However, in spite of the enormous developments in technology and social media over the past 30 years, the essential theme of my book remains intact. The essential theme is that mathematics can be a stimulating, challenging, and thoroughly enjoyable recreational mental activity to enhance and enrich substantial and creative thinking for children in our school system. As I describe in the Appendix, I have experienced a wide variety of appreciative and enthusiastic responses to my Numberama program over the years, ranging across teacher workshops, teacher education programs, children in gifted programs, children in regular classes, liberal arts college instruction, and family math workshops. I have also effectively utilized my Numberama program at a mental hospital for children, a senior citizen center, as a supplement in my algebra classes, and as an example of creative thinking in my psychology classes. Most recently,
I found myself giving a “perfect number lesson” on a napkin at a restaurant at a spiritual development workshop I attended. I had the workshop presenter and some of the participants and staff enraptured,
and I realized that Numberama is deeply ingrained in me, wherever I go and whatever I do.
I am still a pure mathematician, and I practice what I preach. Doing mathematics for me is refreshing, stimulating, meditative, and enjoyable. I occasionally make use of technology to try to find examples of some of my pure mathematics algebraic number theory results, but this is always very secondary, as the priority is on my “thinking.” And this is the exact same philosophy I promote in my Numberama program in regard to the use of technology. Technology is a wonderful tool, but it is essential for the human being to be in control of the technology and not the other way around. Thus finding interesting, surprising, and enticing patterns of numbers, with the assistance of arithmetic calculators when the numbers invariable get very large, is a natural part of my Numberama problems. But what is most important here is the discovery of the patterns themselves, using technology to enhance the discovery.

With this in mind, I am excited to now be offering my *Numberama* book as an e-book through Bentham publications. I welcome feedback from anyone who is using my Numberama problems and games, and I hope that I have succeeded in transmitting the joys of searching for captivating patterns of numbers in my book.

**Elliot Benjamin**

Instructor of Mathematics at CAL Campus,

Psychology Mentor/Ph.D Committee Chair at Capella University,

Minneapolis,

USA

#### List of Contributors

##### Author(s):

**Elliot Benjamin **