# Recommended Books, Games, & Gadgets

We are truly living in a golden age of mathematics books and materials that sneak out of the ivory tower. However, this wealth of goods can make it difficult to find the right one. There are also quite a few out there of mediocre or even poor quality. We hope this list helps narrow down the options. Each item in our list has comments from Epsilon Camp faculty past and present. See what piques your interest, or just choose some at random. There are no duds here.

Books marked with an asterisk* should be accessible even for those who are not quite so excited about math.

There is one thing we would recommend to all:

*** Martin Gardner's Books****Arkady**: Martin Gardner never studied math at college (he majored in philosophy) but did more to stimulate an appreciation for deep and beautiful mathematical ideas in the US then all mathematics professionals combined. His books are full of wonderful non-technical mathematics presented with contagious enthusiasm and love.**Edmund**: Martin Gardner's books are full of a deep love for humanity, curiosity and play. It is hard not to fall into hyperbole when describing them!**Rolfe**: Well, just get anything by Martin Gardner. His books are full of fun and puzzling ideas.

## Puzzles and Fun

** The Big Book of Brain Games** by Ivan Moscovich

**Rolfe**: Not just any "brain games" here, the challenges in this book have real mathematical content. If you're looking for a good source of problems that you'll have to wrestle with to solve, this book is a great alternative to competition Math.

* The Moscow Puzzles: 359 Mathematical Recreations* by B. Kordemsky

**Arkady**: An excellent collection of fun math puzzles of various difficulty for all ages.

**Jeremy**also recommended this as his book to the camp.

* Mathematical Puzzles: A connoisseur's collection* by Peter Winkler

**Martin**: Great puzzles, with entertaining narrative.

* * The Number Devil* by Hans Magnus Enzensberger

**Tara**: This mathematical storybook is pure fun.

* The Man Who Counted* by Malba Tahan

**Tara**: This is in a similar vein to The Number Devil, but may be one that people are somewhat less familiar with.

* The Math Factor* podcast

**Chaim**: There have been no new segments since April 2012, but there are about 200 short pieces on mathematics and puzzles.

**www.setgame.com ****Martin**: Includes many educational resources associated with the game SET.

**Rubik's Cubes****Martin**: Start with *Algorithms and the Rubik's Cube*, a talk by a mathematician. Other resources explain group-theoretical concepts of permutations, conjugates and commutators; preservation of parity and other invariants, and other topics. (Still looking for the best resource to recommend . . . it's out there somewhere.)

## History and Culture

* * Mathematics, A Very Short Introduction* by Tim Gowers

**Edmund**: Fitting the title this is a short, elegant book about mathematics and mathematical thought. The author Tim Gowers is a first rate mathematician (Fields Medal winner no less) but is also very strong at communication, so this is both a clear presentation of ideas and comes from the top of the subject. I give this to anyone who asks about what I do.

* What Is Mathematics? (An Elementary Approach to Ideas and Methods)* by R Courant and H Robbins

**Arkady**: A great 1941 classic, still without rivals. It presents fundamental mathematical ideas through carefully selected examples from number theory, geometry, topology and calculus. Many mathematicians credit this book with opening their eyes and changing their lives.

* Googols, Fractals, and Other Mathematical tales* by Theoni Pappas.

**Rolfe**: Offers a fun tour of some important ideas and problems in Mathematics. Theoni Pappas has written quite a few books, and they are all worth a look.

* * Here’s looking at Euclid* by Alex Bellos

*by Alex Bellos*

*** The Grapes of Math****Edmund:**Two books that describe the culture of mathematics and mathematics in culture, from tribes in the Amazon with only the words one, two, few, many for number and abacus schools to catenary curves and Cellular automata, even the strange world of people’s favorite number.

* * Mathematics: From the Birth of Numbers* by Jan Gullberg

**Arkady:**The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive, profusely illustrated survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

* * The Math Book* by Clifford A. Pickover.

**Rolfe:**This offers interesting 1-page descriptions of 250 important milestones in the development of Mathematics. On its own it provides a nice perspective on the growth and breadth of the field, but it also serves as a nice “shopping list” for ideas you might want to learn more about.

* Men of Mathematics* by E.T. Bell

**Arkady:**A collection of entertaining, fascinating and opinionated stories, legends and myths about mathematics and mathematicians. Many professional and amateur math lovers credit this book for inspiring them and showing what math is and what mathematicians are and do.

**Edmund:**A great story book, but regard all stories as legends (based in fact) rather than history!

## Proof and Logic

**Raymond Smullyan's Books ****Chaim:** Well, these were already mentioned, but I don't mind at all re-emphasizing everything by Martin Gardner and Raymond Smullyan's puzzle books, such as The Lady or The Tiger, This Book has no Title, or What is the Name of This Book (and several others) Logical Labyrinths is a good next step. **Tara:** Many kids who enjoy books by Martin Gardner will also enjoy some of Raymond Smullyan's logic puzzle books.

* Conjecture and Proof* by Miklos Laczkovich.

**Rolfe:**This book offers mini-introductions to various areas of Mathematics including topics in Geometry, Topology, and Number Theory. Along the way, it provides accessible but rigorous proofs of some deep results.

* Surreal Numbers* by Donald Knuth

**Tara:**I remember enjoying Surreal Numbers by D.E. Knuth when I was younger also -- another one of the mathematical storybook genre.

**Edmund:**Follow a young couple escaping from the world on a beach in India as they explore new worlds of numbers and learn what being a mathematician means.

**Roger Nelson’s Books ****Tara:** Roger Nelsen has written or co-authored several fun books -- Proofs Without Words, Charming Proofs, etc. that give nice proofs of elementary results and would be accessible to stronger students I think.

* Proofs From the Book* by M Aigner, G M Ziegler and K H Hofmann

**Martin:**Erdös said that God has a Book of the best (most elegant or inspiring) theorem for each problem. This volume contains some of them. (Possibly wait for an upcoming newer edition.)

## Problem Solving

* Crossing the River with Dogs* by Ted Herr, Ken Johnson and Dan Piraro

**Tara**: This is a nice introduction to problem solving.

* For All Practical Purposes: Mathematical Literacy in Today's World* (COMAP)

**Tara:**This provides an accessible introduction to a number of topics not part of the standard curriculum (graph theory, voting schemes, and so forth).

* Mathematical Circle Diaries, Year 1: Complete Curriculum for Grades 5 to 7* by Anna Burago

**Arkady**: The book contains detailed well-organized math circle lessons with problems, solutions and pedagogical advice, enough for a group meeting once a week to occupy themselves for a whole school year.

* A Moscow Math Circle: Week-by-Week Problem Sets* by Sergei Dorichenko

**Arkady:**A collection of fun and thought-provoking problems (with complete solutions) used during one year at the math circle for middle-school students in Moscow.

* Mathematical Circles: Russian Experience* by D.Fomin, S.Genkin, I.Itenberg

**Arkady**: This book is an excellent source for both teachers and students of problem-solving techniques. It has a very good collection of problems organized by topics (parity, the pigeonhole principle, divisibility, induction, combinatorics, invariants, graphs and trees, etc) with each chapter progressing from an introductory remarks and worked examples to quite non-trivial problems.

**Rolfe**: This book offers wonderful series of progressively more challenging problems in number theory, combinatorics, graph theory, and more.

* Math Olympiad Contest Problems for Elementary and Middle Schools* by George Lenchner

**Rolfe**: These books have problems (with solutions) for past MOEMS contests. The problems are a great way to develop logical and algebraic thinking without demanding significant prerequisites.

* Creative Problem Solving in School Mathematics *by George Lenchner.

**Rolfe**: Not a very well organized book, but it offers a toolkit of general problem solving strategies, a tour of different common problem types, and a nice collection of practice problems.

* The Art and Craft of Problem Solving* by Paul Zeitz

**Arkady**: An excellent book both for experienced problem-solvers and serious beginners. It contains many well-selected problems (most without solutions though).

* Art of Problem Solving* (Books and Courses)

**Tara**: Most parents will know about Art of Problem Solving -- if they didn't before camp, they will now. (Edmund) More general than most of this section, but with that name I had to include it here!

## Geometry, Origami and Topology

* Kiselev's Geometry: Book I. Planimetry *and

*by A.P. Kisilev*

**Kiselev's Geometry: Book II. Stereometry**Adapted to the US curricula and with supplements by A Givental, a UC-Berkeley math professor

**Arkady**: Simply the best Euclidean Geometry textbook available (and by far the most popular Russian textbook, in all subjects for all times). First published in 1892, it has been published over 40 times in dozens of millions of copies, and lived through many epochs, wars, reforms and revolutions (not only in education). The volumes are relatively small, but due to Kiselev's concise style, all topics are given comprehensive treatment with no cut corners.

* Geometry Revisited* by H.S.M. Coxeter, S.L.Greitzer

**Arkady**: A very good textbook on "advanced" plane geometry, a perfect continuation for a motivated student after a standard high school course. It contains many beautiful and nontrivial theorems (e.g. those of Ceva, Menelaus, Pappus, Desargues, Ptolemy, Pascal, Brianchon, and Morley) and emphasizes the use of transformations (including inversions and projective transformations).

**Two Ruler and Compass Games** **Chaim**: Two excellent compass and straightedge construction "games" are at http://sciencevsmagic.net and

**Geometer’s Sketchpad ****Chaim**: This program has an enormous base of resources available. It is robust and wonderful.

**Graphing Calculator ****Chaim**: Not just a graphing calculator, but a poorly named, very interesting piece of software. It is very flexible and worth exploring. (The story of its development, at http://pacifict.com/story is pretty remarkable in its own right) Check out the demos and examples. This program is really an understated gem, and I very highly recommend it.

* Zome Geometry* by George W. Hart and Henri Piccioto.

**Rolfe**: This book provides a series of hands-on activities combined with pencil-and-paper questions that will give a solid understanding of some fundamental results in 2 and 3D geometry (with some 4D stuff thrown in). As a bonus, the lessons give the reader a solid understanding of how to use the Zome system.

* How to Fold It* by Joseph O’Rourke.

**Rolfe**: An accessible but still serious book about the mathematics of origami, linkages, and more.

* Polyhedron Models* by Magnus J. Wenninger.

**Rolfe**: With plenty of pictures of complex polyhedra, this is a fun book to flip through. If you start reading, it gets even better as you start to understand the subtleties of what you’re looking at. You might even find yourself cutting up some paper and building them yourself.

* The 59 Icosahedra* by H S M Coxeter, Du Val, Flather, and Petrie.

**Rolfe**: More focused than “Polyhedron Models”, this book is still full of interesting pictures, but also offers a great example of systematic analysis of a geometric problem.

* Project Origami* by Thomas Hull

**Martin**: A variety of paper folding activities in many branches of mathematics, including curvature, topology, number theory, and combinatorics. Inspired by traditional origami and will appeal to origami fans, but does not give traditional origami instructions.

* Geometry and the Imagination* by David Hilbert and S. Cohn-Vossen

**Arkady:**Another masterpiece of mathematical exposition written by one of the leading mathematicians of the 20th century. The book describes beautiful ideas and examples from various topics in Geometry (projective geometry, conic sections, regular polyhedra in 3 and 4 dimensions, lattices and crystallographic groups, non-Euclidean geometries, topology of surfaces, Gaussian curvature, etc).

* iOrnament*, by Jürgen Richter

**Chaim**: This is a very nice symmetry program for the iPhone and iPad.

* * Crocheting Adventures in Hyperbolic Planes* by D Taimina

**Edmund**: A satisfyingly hands on approach to Hyperbolic Geometry.

* * Euler’s Gem: The Polyhedron Formula and the Birth of Topology* by D Richeson

**Edmund**: A readable and fast paced account of the history, and central ideas of topology, built around the great insight of the importance of the Euler Characteristic.

* The Knot Book* by Colin Adams

**Chaim**: A very good, very readable introduction to Knot Theory, which is closely entwined with the study of three-manifolds.

**Cornelia:**This was already on Chaim's list, but I just wanted to chime in that I agree!

* Experiments in Topology* by Barr

**Chaim**: This has some paper constructions for exploring surfaces.

* Three Manifold Topology and Geometry Vol 1* by Field's Medalist William Thurston

(the only volume, sadly)

**Chaim**: This will be way over their heads, but some of the students will find it stimulating in any case, and it really shows off where this subject goes.

## Basic Math and Algebra

* Basic Mathematics* by Serge Lang

**Arkady**: A famous mathematician Serge Lang presents the topics that he feels a well-prepared student ought to know before starting calculus. It's not an "exciting" book to read, but it deals with the basic math topics in a serious, non-condescending way and can serve as a good review or reference text.

### Gelfand's school books

* Algebra* by I.M. Gelfand, Alexander Shen

*by I.M. Gelfand, E.G. Glagoleva, E. E. Shnol (Dover)*

**Functions and Graphs***by I.M. Gelfand, E.G. Glagoleva, A.A. Kirillov (Dover)*

**The Method of Coordinates***by I.M. Gelfand and Mark Saul*

**Trigonometry****Arkady**: I.M. Gelfand, one the 20th century's most renowned mathematicians, co-wrote a series of books for his Correspondence Math School in Russia. These small books provide clear and lively treatment of the standard high school material with emphasis on development of the ideas and connection between different areas of mathematics. Books are written with higher mathematics in mind and often venture into topics usually not covered in the high school curriculum. There are many exercises which are well-chosen, thought-provoking and sometimes quite challenging. Even students already familiar with the material will benefit from reading the books.

The first of these books, Algebra, should be a required reading for all epsilon-campers. The last three present a decent substitute for a high school precalculus course.

**Rolfe** (on Algebra): Concise, fun to read, and with problems that lead you deeper, this book is a great way to learn or review Algebra.

* E-Z Algebra* by Douglas Downing

**Rolfe**: Yes, it’s a Barron’s guide, but Downing’s books are fun to read and have characters that ask the “stupid questions” some of us are too afraid to ask. Not for everyone, but some kids really like these books.

* Algebra* by Michael Artin

**Arkady**: A very good self-contained introduction to abstract algebra and its applications. It includes linear algebra as well, so the only prerequisite is some "mathematical maturity" (i.e. being comfortable with abstract concepts and understanding and writing proofs).

* Visual Group Theory* by N Carter

**Tara**: A reasonable follow-up book for what I did with the third years (it's an MAA book and pretty inexpensive as an eBook; the author also has some on-line material I think, and there is a free software package to explore finite groups as well).

## Other Mathematics

* Algorithms Unplugged* by Various

**Martin**: Algorithms is generally regarded as a discrete branch of mathematics—algorithms are proven to be correct and efficient, or an invariant is found for a problem to prove that there can be no such algorithm. This book gives a sampling from many algorithmic topics. Written colloquially for high school students, it has many elegant proofs and no dependence on programming languages or computers (though there are some pointers to computer-based followups on the web).

* Calculus (4th edition)* by Michael Spivak

**Arkady**: The canonical choice of a Calculus textbook for any bright mathematically-minded (high school or college) student. Written in a clear and entertaining style with a multitude of carefully selected exercises, this book presents Calculus from scratch, with numerous (often non-trivial) applications.

* Concrete Mathematics: A Foundation for Computer Science* by R.L.Graham, D.E.Knuth and O.Patashnik

**Arkady**: A remarkable textbook on Discrete Mathematics (understood here as math needed for theoretical computer science: recurrence, sums, number theory, combinatorics, generating functions, etc) with a very good selection of material and tons of concrete examples and solved problems. Beautifully written in a clear fun-spirited style with elegant proofs, it is a serious book which requires serious thinking.

**Desmos****Edmund**: A thoughtfully designed and elegant graphing calculator, it takes some effort to make it give an error. Can graph, make sliders and animate giving an excellent way to explore functions and more visual mathematics. The seemingly simple system is surprisingly powerful. I have some notes here: http://maxwelldemon.com/2013/08/03/form-follows-functions/

## Other Books for Parents and Beyond

These books should be accessible even for those who are not quite so excited about math.

* * Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers* by Alexander Zvonkin

**Arkady**: A captivating and well-written account of an attempt by a professional mathematician to conduct a math circle for children between 3 and 6. In addition to many fun math-related activities, the book is full of interesting psychological and pedagogical observations.

* * Arithmetic for Parents: A Book for Grownups about Children's Mathematics* by Ron Aharoni

**Arkady**: This book is written by a professional mathematician who once accepted a challenge to teach in an elementary school and now wants to share his experience which turned out to be truly educational and eye-opening. The first half of the book, where Aharoni talks about the nature of mathematics, the role of abstraction, and the principles of teaching it, would be interesting not only for parents of elementary school kids.

* * A Mathematician's Lament* by Paul Lockhart

**Arkady**: One of the best critiques of the current state of math education written with eloquence and passion. The Lamentation part continues with Exultation, where the author presents his view on what mathematics really is about (the purest art form, of course). This small book is a must read for parents, math educators and politicians.

* * Arithmetic for Parents: A Book for Grownups about Children's Mathematics* by Ron Aharoni

**Arkady**: This book is written by a professional mathematician who once accepted a challenge to teach in an elementary school and now wants to share his experience which turned out to be truly educational and eye-opening. The first half of the book, where Aharoni talks about the nature of mathematics, the role of abstraction, and the principles of teaching it, would be interesting not only for parents of elementary school kids.