Uncle Petros and Goldbach’s Conjecture, by Apostolos Doxiadis
Uncle Petros is a recluse. Our hero, his nephew, is trying to discover his secrets. It seems he was close to proving Goldbach’s conjecture, that every even number greater than 2 is the sum of two prime numbers. There is just a tiny bit of math in this, but lots of (slightly twisted) history of math. (Some mature content.)

Journey Through Genius: The Great Theorems of Mathematics, by William Dunham
Each chapter is about both the theorem and the mathematicians who proved it. There is a great chapter on the history behind finding a formula to solve cubic equations, which led to the creation of complex numbers.

In Code, by Sarah Flannery
Sarah Flannery is the daughter of a mathematician. Because she had an interest in math and science, her teacher encouraged her to enter the Young Scientist Exhibition – a nation-wide competition in Ireland. Her father suggested doing a cryptography project, and she took off with it. After winning the competition in Ireland, she went on to win the corresponding all-Europe competition. She explains her interest in mathematics – with a chalkboard in the kitchen and puzzle discussions over dinner, it came naturally – and the details of her cryptography project.

Mathematical Carnival, by Martin Gardner
There are nineteen short chapters, each from nine to twenty-one pages long. The one on sizes of infinity, a topic I thought I knew well, had a surprising question I hadn’t previously thought about. This is one of dozens of puzzle books put together by a master. Check out any of his books to discover a wealth of math and logic puzzles.

Mathematics: A Human Endeavor, by Harold Jacobs
This one is a textbook, and it’s delightful. The first chapter, on inductive and deductive reasoning, uses pool tables to get the reader thinking about patterns. Chapters on sequences, graphing, large numbers, symmetry, mathematical curves, counting (permutations and combinations), probability, statistics, and topology round out an introduction to a wide variety of math topics, accessible before algebra.

The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel
Ramanujan’s story would be unbelievable if it were fiction or even slightly fictionalized. Ramanujan was too focused on his own mathematical work to do well in school – he was kicked out of college when he failed exams in his other subjects. It took him years of supporting himself working as a clerk before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others.

Chances Are: Adventures in Probability, by Michael and Ellen Kaplan
History, philosophy, science, and statistics all come together in this delightful exploration of probability.

Surreal Numbers, by Donald Knuth (requires well-developed math skills)
This book requires lots of work, doing the math, and what fun work it can be! Alice and Bill are enjoying their extended vacation on an isolated tropical beach, but are getting a bit bored, when they discover a rock with two “rules” on it. Conway has invented a strange number system through these two rules, and Alice and Bill (along with the reader) are sucked in, trying to figure out how it all works. This is higher math.

Measurement, by Paul Lockhart
Paul Lockhart knows that the joy of math comes from figuring things out for ourselves, so he shows us some of his favorite problems and asks us if we’d like to solve them. (He gives lots of hints, strategies, and techniques, and few answers.) But he went beyond a mere compendium of puzzles, and connected the problems he shows, taking his readers on a delightful journey though size and shape (part one) and time and space (part two). This book is a delight, even if you’re not feeling up to solving many of the puzzles. But the more often you put down the book and pick up your pen or pencil and paper, the more fun you’ll have.

Mathsemantics, by Edward MacNeal
This book has one great chapter on estimation that’s worth getting the whole book. He talks about having a semantic web in your head that includes a few important numbers, like: the radius of the earth and the populations of the earth, your country, your state, and one very large city. He recommends using these as a basis to estimate often, committing to your estimates somehow, and then finding out the real values of what you estimate. For example, estimate your arrival time when you’re in the car, tell the person next to you, and notice the time when you do arrive at your destination. You’ll see yourself getting better and better at estimation, along with strengthening your number sense.

Euclid in the Rainforest, by Joseph Mazur
Logic, infinity, and probability are the topics. Adventures in Venezuela, Greece, and New York furnish the background. Mazur has wide-ranging interests, and skillfully brings the math to life.

The World of Mathematics, by James Newman
This four-volume set is a mathematical encyclopedia – perfect for browsing. The articles are approachable and intriguing. My favorite so far is On Being the Right Size, by J.B.S. Haldane (now available online), in which the author explains why giant insects are not possible (at least on Earth), nor people-shaped giants. Why? Because surface area grows as the square of height while volume grows as the cube of height, and systems get out of balance when the ratio of surface area to volume changes dramatically.

How to Solve It, by George Polya
Written in 1945, this book is so good that most math textbooks that discuss problem solving just paraphrase Polya. It has a great summary in front, but the organization of the rest of the book is rather strange – it’s alphabetical by the first word of the idea being discussed.

What Is the Name of this Book?, by Raymond Smullyan
Knaves always lie and Knights always tell the truth. Puzzle number 29: A says, “Either I am a Knave or B is a Knight.” What are A and B? This mysteriously-named book has 271 twisted logic puzzles.

The Joy of X: A Guided Tour of Math, from One to Infinity, by Steven Strogatz
He’s not talking down to the reader, and yet this mathematician makes it all clear, from numbers to topology and calculus. He started out writing columns on math in the New York Times, and that morphed into this delightful book with thirty short chapters that can each be read independently.

Who Is Fourier?, by Transnational College of LEX
Fourier Series are used to describe sound. Usually an advanced college topic, they are explained in unique, easy-to-understand ways in this charming book, accessible to anyone who has a basic understanding of algebra.

Math Girls and Math Girls 2, by Hiroshi Yuki
The unnamed protagonist is a boy in high school who loves math. He helps Tetra with her math, and is challenged by the problems Miruka poses. In Math Girls 2, a few more girls join the gang. The math is challenging in these books, and the storyline makes it all the more fun.

The Art and Craft of Problem-Solving, by Paul Zeitz
How do you go about solving challenging problems? Zeitz discusses tools, tactics, and strategies, and offers a rich storehouse of very challenging problems.