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
The Grapes of Math by Alex Bellos
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 Kiselev's Geometry: Book II. Stereometry by A.P. Kisilev
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
Functions and Graphs by I.M. Gelfand, E.G. Glagoleva, E. E. Shnol (Dover) 
The Method of Coordinates by I.M. Gelfand, E.G. Glagoleva, A.A. Kirillov (Dover) 
Trigonometry by I.M. Gelfand and Mark Saul
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.