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My friends have periodically asked me to recommend books that could help advanced children improve their mathematics. So, I have pulled together this resource list of particularly good math books for kids, along with my comments on why they are so good. I've linked as many of these as I could to their pages on Amazon.com. As a note, those links take you to Amazon.com, where you will be subject to their policies and procedures. I take no responsibility for what happens to you over there :)
Since I work with very advanced children, you need to take that into account when you see my age recommendations.
For kindergarten and elementary school, and probably some adults "Imagine" by Norman Messenger has beautiful pictures, each one a puzzle, joke or optical illusion. As a bonus, at the corners of every page there is a math puzzle for older children. 

For kids, starting from kindergarten, and for adults While "The Book of Classic Board Games" by Klutz Press is not exactly a math book, it would be a most excellent possession. This book details 15 board games, including their complete rules and, more importantly, boards and a set of pieces for all of them. So one can play all the games contained in it right away. The games in this book are classics. They have all stood the test of time, most having been popular for centuries, and some even for millennia. All these games are easy to learn, despite having deep (sometimes very deep) strategies. Both children and adults will enjoy this book. 

For kids, starting from kindergarten, and for adults "The Magic Show" by Mark Setteducati has the unique (as far as I know) property of being a selfpresenting magic show. The book has various tabs and wheels that one pulls or rotates to make things happen. The interesting part is that the tricks tend not to involve subterfuge, but rather are mathematical in nature. The book is not only fun: figuring out how the tricks work is also educational. 

For kids in elementary and middle school The name of "Cool Math" by Christy Maganzini is apt — the math presented here really is cool. It is appropriate for children slightly past the stage of being able to count to a million and to multiply. I used to teach at School Plus, and later discovered that the contents of this book coincided precisely with my program. That is, the author of this book and I have the same sense of coolness. This book constitutes a children's version of MathAlive, a math class at Princeton University that is also cool. 

For kids starting middle school, and for adults "A SuperSneaky, DoubleCrossing, Up, Down, Round & Round Maze Book" by Larry Evans is an excellent book of mazes. Each maze differs from the others, and each one is a work of art. 

Grades 58 I conducted a Math Party for the benefit of a Princeton Charter School fundraising auction. "Solve This" by James S. Tanton was the best book for it. It's particularly appropriate for kids who do Mathcounts. Solve This is a book of activities — rather than exercises — that work well for a group effort or for an individual who wants to show off to their friends. 
These books can be enjoyed by kids and adults at the same time.
For kids and adults "Masters of Deception" by Al Seckel is the most mathematical art book I've ever seen. As you would expect, this book has chapters with paintings by Dali and Escher. But it also includes ambigrams by Scott Kim, anamorphoses by Istvan Orosz and many other artists who use optical illusions and impossible figures. The day I brought this book home, my family cancelled all our plans because we simply could not put this book aside until we finished it. I like art when it is not only beautiful, but when it also puzzles me. 

Beginning of the book: 8 and up. End of the book: 20 and up "The Riddle of Scheherazade" by Raymond Smullyan is a fascinating book of logic puzzles. Unlike some puzzle books, the puzzles in this one build in difficulty. "What is the Name of this Book?" by Raymond Smullyan is likewise structured with a progression of logic puzzles, running from the classic "You have two coins with total value 15c. One of them is not a nickel. What are they?" to Godel's Incompleteness Theorem. Raymond Smullyan is, in my opinion, the best author of logic books of all time. He totally knows his stuff, and has an exceptional sense of humor. He has written a great many books on logic and on the Tao. His perspective on the latter is very interesting because it is the perspective of a logician. Feel free to search through Amazon for Raymond Smullyan. 

Age 9 and up "Aha! Gotcha" by Martin Gardner is about paradoxes. It presents numerous paradoxes, complete with amusing pictures, and then dissects and discusses each one in nontechnical terms. It's great fun if you aren't acquainted with paradoxes, and potentially enlightening even if you are. My 11yearold son has read this book more than ten times. Martin Gardner is the classic author of recreational mathematics books. He has written a huge amount of stuff — brainteasers, hexaflexagons, discussions of the structure of the universe — you name it. One particularly good example is "Classic Brainteasers" (on the right). Go ahead and search Amazon for Martin Gardner. You're likely to find something to your taste. 

Ages 9 and up "Logic puzzles" by Mark Fowler has a delightful collection of logic puzzles, which, when taken together, contain the clues to a "superpuzzle." The "superpuzzle" provides a unifying thread that motivates the kids to persist in solving the individual puzzles in the book. 

For kids starting from middle school, and for adults "Walter Wick's Optical Tricks" by Walter Wick is a book of optical illusions. The fascinating thing about this book, though, is that all the illusions are presented as photographs. There are pictures (real, no photo editing!) of impossible shapes, objects floating in midair, and other fascinating optical tricks. The book encourages the reader to figure out how each photograph was taken. Try it! You might recognize Walter Wick's spectacular photography from his his photowork in the "I spy" series. 

For nonbeginner chess players "The Chess Mysteries of Sherlock Holmes" by Raymond M. Smullyan is not your typical chess puzzle book. Instead of asking you to find a mate in three steps or to win material, this book offers entirely different challenges. For example, it presents a position, tells you that it was obtained legally, and then asks you to find the last two moves. Sometimes it gives you a position with one piece missing and asks you to place it (again, assuming legality throughout the game). After engaging with this book, you'll see chess in a thoroughly altered light. "The Chess Mysteries of the Arabian Knights" by Raymond M. Smullyan is similar to the previous book, but the puzzles are more difficult. 
Here are fun math books for adults.
For people interested in Greek myths In the book "Mythematics: Solving the Twelve Labors of Hercules" Michael Huber adds details to Hercules' labors so that in order that he can do each task, you need to help Hercules solve two or three math problems. For example, to defeat the Nemean Lion Hercules needs to solve the problem "Zeus Makes a Deal", which is a Greekmyth version of the Monty Hall problem. The problems in Mythematics are quite advanced. They range in topic from algebra, geometry and probability to differential equations and integral calculus. Plus, as a reward for helping Hercules, Huber gives you variations on Sudoku puzzles. Solving some nice math problems might not be the only reason for people to buy this book. Here are some other reasons:
I like Huber's approach. Future possibilities for more books are endless. Let's write new math problems based on Harry Potter, Batman, the Bible or, maybe, The Joy of Sex. 

For inteview preparation and for fun I bought the book "Heard on The Street: Quantitative Questions from Wall Street Job Interviews" by Timothy Falcon Crack several years ago when I was looking for a job and felt that working in finance was a possibility. Despite having bought it simply to prepare for employment interviews, I actually enjoyed the math problems in the book. The book has problems in logic, probability, statistics and finance, as well as a very useful chapter of general interview questions. If you're interested in buying this book, I should mention that some questions require calculus and knowledge of financial terms. I do love the author's taste in problems. 

For adults "The Symmetries of Things" by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss is now published in a beautiful edition with terrific pictures. The first chapter is very nicely written and is suitable for high school and undergraduate students. It covers symmetries of finite and infinite 2D objects. The second chapter adds color to the theory. For colorful pictures with symmetry, there are two symmetry groups: the group that preserves the picture while ignoring its coloring and the group that preserves the picture while respecting its coloring. The latter group is a subgroup of a former group. This second chapter discusses all possible ways to symmetrically color a symmetric 2D picture. The chapter then continues with a discussion of group theory. This chapter is much more difficult to read than the first chapter, as it uses a lot of notations. The pictures are still beautiful, though. The third chapter is even more difficult to read and the notations become even heavier. This chapter discusses hyperbolic groups and symmetries of objects in the hyperbolic space. Then the chapter moves into 3D and 4D. I guess that some parts of the second and the third chapters are not meant for light reading; they should be considered more as reference materials. 

For adults If you love mathematics and the show "Numb3rs", this book is for you. "The Numbers Behind NUMB3RS: Solving Crime with Mathematics" by Keith Devlin and Gary Lorden is a book about the mathematics in the TV series "Numb3rs". The book takes several mathematical situations from the series and discusses the real math behind each one. It gives examples of actual crimes where this math was used. I couldn't stop reading this book. Besides, I was delighted to see the name of my great friend and coauthor Ingrid Daubechies on page 134 discussing wavelets, which are very useful in analyzing fingerprints. 

For adults "PopCo" by Scarlett Thomas is a novel that discusses mathematics. It describes prime numbers, Fibonacci numbers and the Monty Hall problem as part of its narrative. I had a strange feeling reading the math parts of this novel. The descriptions did not seem thorough enough for people who do not know math, but were not needed at all for people who do know math. I have to mention that there are some mistakes. For example, she refers to this statement from a Cretan — "All Cretans are liars" — as a paradox. There is nothing paradoxical about it if the speaker is a liar and not all Cretans are liars. In standard math literature it is usually assumed that all people living on the same island are either liars or truthtellers. In this case, if we assume that, then the above statement becomes a paradox. The author repeatedly cites classic math books and theories without understanding them. Because of that, Scarlett Thomas doesn't mention some implicit assumptions, leading her to make mistakes. There are also arithmetic mistakes. If you don't care about the mathematical details, or if you do care and can see through the mistakes yourself, you might enjoy this book. For me, it was fun to read. 

For adults I always wanted to find mathematical papers which applied mathematical ideas to sex, love and marriage. "Mathematics and Sex" by Clio Cresswell does this. Unfortunately, I strongly dislike a certain element of the book. The author embeds formulae into the text without explaining the meaning of each variable. In my opinion, this is poor mathematical writing. At the same time, these formulae are not crucial to the book and could easily be eliminated. My impression is that the formulae were only included to impress the reader with how complicated the subject looks. Even if the formulae are ignored, the results and goal of each formula are still explained clearly in words. Also, there is a good list of references to actual mathematical papers. 

For adults "The Music of the Primes" by Marcus du Sautoy describes the history of the hunt for a proof of the Riemann Hypothesis. The author is a mathematician and he clearly knows what he is talking about. Furthermore, the second part of the book is dedicated to the modern research of the Riemann Hypothesis, and I personally know some of the people he mentions. This added a personal touch to my experience of reading the book. 

For high school kids preparing for competitions, and for adult math puzzle lovers In "Mathematical Puzzles" Peter Winkler collected those math problems that would best prepare you for the Math Olympiads. To do these problems you don't require math knowledge beyond high school, but you do need inventiveness and creativity. Peter has outstanding taste, so his collection contains marvelous math puzzles. "Mathematical MindBenders" is another book by Peter Winkler written in the same style. 

For people preparing for interviews and adult math puzzle lovers "How Would You Move Mount Fuji? Microsoft's Cult of the Puzzle  How the World's Smartest Company Selects the Most Creative Thinkers" by William Poundstone is more than a book of puzzles: it is also like a book of gossip. Poundstone gives readers some fun puzzles taken from Microsoft interview questions, but the book also describes some interviews and discusses the culture of interviewing creative thinkers. The book includes the most famous Microsoft question — one I wrote about on my own blog: "Why are manhole covers round?" 
I was on the Soviet IMO team twice and now my younger son is preparing for USAMOs. Here is my list of useful literature.
In "Mathematical Puzzles" Peter Winkler collected those math problems that would best prepare you for the Math Olympiads. To do these problems you don't require math knowledge beyond high school, but you do need inventiveness and creativity. Peter has outstanding taste, so his collection contains marvelous math puzzles. "Mathematical MindBenders" is another book by Peter Winkler written in the same style. 

The book ";International Mathematical Olympiad 19591999" by Istvan Reiman contains all the problems from the first 40 IMOs together with their solutions. It is a must for every student preparing for IMO. Reiman sometimes provides two different solutions, and in rare cases — three or four. He also discusses some generalizations. 

I like how "The Art and Craft of Problem Solving" by Paul Zeitz is structured. The problems are grouped around ideas and strategies. First Zeitz presents an idea or a strategy, then he dissects some examples. The book includes a lot of problems and exercises. There are no solutions — only hints about selected problems. For those of us who are impatient and might succumb to the urge to check the solution, there's a real advantage to working with at least one book with a lot of problems and no solutions. 

"The USSR Olympiad Problem Book" was the Olympiad Bible when I was in high school. The legend was that if you were allowed to have only one book to prepare, then this was the one. This book restricts itself to only arithmetic and algebra problems. It has 320 mostly classical problems and it covers all the arithmetic and algebra Olympiad ideas. 

"Mathematical Olympiad Challenges" by Titu Andreescu is organized by ideas. For each idea there are several examples followed by the problems and ending with the solutions. This book is similar in its structure to Paul Zeitz's book, but more advanced. The downside of this volume is that the writing reflects the fact that the author is not a native English speaker. 

The standard geometry course in the US is insufficient preparation for math competitions. If you want to win, you have to devote a big part of your time to geometry. "Lines and Curves: A Practical Geometry Handbook" by Victor Gutenmacher and N.B. Vasilyev is a very special geometry book: it presents geometry in motion. This unusual, dynamic approach is very engaging. The explanations are interlaced with numerous, tastefullychosen problems of increasing difficulty. 

"Geometry Revisited" by H. S. M. Coxeter and Samuel L. Greitzer is a fun book. It has a section for every important geometry theorem and type of problem you're likely to find in the competitions. For example, it covers the ninepoint circle theorem and "The Butterfly" problem. 
Last revised February 2009