An aggressively unsimplified account of 14 great problems, emphasizing how mathematicians approached but did not always solve them.

Fermat’s Last Theorem, 350 years old and solved by Andrew Wiles in 1995, produced headlines because laymen were amazed that mathematicians could make new discoveries. In fact, mathematics is as creative as physics, writes prolific popularizer Stewart (Mathematics Emeritus/Univ. of Warwick; *The Mathematics of Life*, 2011): “Mathematics is newer, and more diverse, than most of us imagine.” Goldbach’s Conjecture—that every even number can be written as the sum of two prime numbers (250 years old, probably true but not proven)—provides the background for a chapter on the unruly field of prime numbers: those divisible only by one and itself (3, 5, 7, 11, 13…). Squaring the Circle—constructing a square with an area identical to a given circle (2,500 years old; proven impossible)—introduces pi. Schoolchildren learn that pi is the ratio of the circumference of a circle to its diameter, but it’s a deeply important number that turns up everywhere in mathematics. Most readers know that Newton’s laws precisely predict motions of two bodies, but few know that they flop with three. The Three-body Problem (330 years old, unsolved) continues to worry astronomers since it hints that gravitational forces among three or more bodies may be unstable, so the planets may eventually fly off.

Stewart’s imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating many but not all of some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention.