Mathematical problems mostly come in two kinds. There are the old and easy ones, like “What do you get when you multiply six by seven?,” whose answers can only excite seven-year-olds and autistic adults. Then there are the problems that interest research mathematicians, whose incomprehensibility begins to fade only after several university degrees in the higher algebra. It is hard to make dinner-party conversation or requests for funding out of a discipline whose very terms are well beyond the reach of ordinary mortals. There are a few mathematical questions that can be understood by anyone but whose solutions have been achieved only by the last few decades of difficult research. John Casti’s book presents five of them.

Sir Walter Raleigh asked his mathematical adviser for a formula for the number of cannonballs in various stacks on the deck of his ship. The adviser came up with something, but realized it was not clear whether the balls were being stacked in the most efficient way. There is a natural way to pack spheres tightly: lay down the deck layer in “hexagonal” fashion, where each ball is surrounded by six others. Then put the second layer so that the balls lie in every second depression in the first layer; the second layer will then also form a hexagonal packing. Then put the third layer in every second depression of the second layer, so that the balls of the third layer are directly above those of the first. If this is done on a large scale, the balls will occupy π/√18 of the space, or just over 74 percent. Is this the densest packing possible? Kepler conjectured that it was. Fruit shop owners thought it was obvious. Casti reports a New Zealand greengrocer amazed to hear that mathematicians had been working on the problem for four hundred years. Asked how long it took him to find the best packing, he said “You just put one on top of the other. It took me about two seconds.” But conjecture and tradition are not enough for mathematicians. They demand proof. A proof now exists, but it was not achieved until 1998, when Thomas Hales of the University of Michigan put together the last steps, using computer verifications of many complicated cases.

Casti’s other four problems are also easy to grasp, though some more so than others. Perhaps easiest is the Four Color Theorem. Maps are colored so that countries sharing a border have different colors; the theorem says that no matter how complicated the map is, no more than four colors are ever needed. (That is, for maps on a plane or sphere—for maps on the surface of a donut, it is different, and one of Casti’s color plates shows a map on a donut that requires seven colors.) The appeal of the result lies in its simplicity, which underlies the potentially infinite complexity of actual maps. It was conjectured in the mid-nineteenth century, but only proved by Appel and Haken in 1976. Their proof used a good deal of computer verification of cases (checked by another computer), leading to some soul-searching in the mathematical community over whether the old ideal of simple, understandable, and surveyable proofs was dead. Would mathematicians be replaced by machines, as clerks with pencil and paper had been replaced by calculators?

The self-image of humans received a boost with the solution of the next of Casti’s five problems, Fermat’s Last Theorem. Conjectured by Fermat around 1650, it says we can, for example, be sure the equation 85 + 135 = 145 is not true, without doing any calculations. For no equation of the form xn + yn= zn is true, for any positive whole numbers x, y, z, and n (except for n=1, obviously, and n=2, when some cases such as 32 + 42 = 52). Three-and-a-half centuries of hard work later, the theorem was proved by Andrew Wiles and Richard Taylor in 1994 with no assistance from a computer.

The fourth problem is the Continuum Hypothesis, which concerns the comparability in size of infinite sets (some infinite sets are bigger than others, and it is unclear which is the second smallest). The last is Hilbert’s Tenth Problem, that is, the tenth in David Hilbert’s list of twenty-three major unsolved problems in mathematics, which he set in 1900 as a challenge for the new century. It asks whether there is a mechanical procedure, such as could be programmed into a computer, for deciding whether a Diophantine equation has solutions (a Diophantine equation is one like the Fermat equation xn + yn = zn, where solutions must be whole numbers). The Leningrad mathematician Matyasevich proved in 1970 that the answer is in the negative: no such procedure exists. (Soviet pure mathematics was always very strong: geniuses gravitated towards it because it had no political problems and no need for computers.) Hilbert was thus shown to have had a double prescience: firstly, in understanding the importance of problems about what can be computed fifty years before computers were invented and secondly in choosing a problem that revealed a fundamental limitation to what computers can do.

The crucial question for a popular book on mathematics is whether the average reader is going to understand it. In this case, the answer is sometimes yes, sometimes no. Unfortunately, it is not always clear what the reader is expected to grasp. At one moment, rational numbers will be defined, then a few pages later the reader will be expected to follow logically complex arguments like “every square is either a multiple of 4 or one more than a multiple of 4, depending on whether it is the square of an even or an odd number. So the difference of two squares can never be two more than a multiple of 4 …” There are also some minor errors that impede understanding. But if the reader is prepared to plow on without worrying about not understanding some sentences, there is genuine insight to be had. Things are a lot clearer, certainly, than in books on popular cosmology and quantum physics. And even if much in the hard parts of the proofs inevitably cannot be explained, the questions themselves are properly laid out, along with why they are important and at least some idea of the concepts in the proofs.

The book is especially good on some of the related problems that the work on these “Mathematical Mountaintops” has led to. The survey of yet unproved conjectures that generalize Fermat’s Last Theorem is excellent, as is the material on foams. The bubbles in foams, like the spheres of Kepler’s Conjecture, fill space with an array of regular shapes. Among the strangest mathematical entities discovered in the 1990s is the Weaire-Phelan foam, which fills space remarkably efficiently by means of two types of cells, one with fourteen sides and one with twelve. The reader inspired to find a book where it is possible really to understand everything might try Keith Devlin’s Mathematics: The Science of Patterns.

Mathematics is not running out of problems. Casti promises another book on some of those still unsolved, including the one now regarded as the Everest of the mathematical world, the Riemann Hypothesis (it is not easy to explain what this is). He lists in an appendix the seven Clay Prize problems, for whose solution the Clay Mathematics Institute has offered a million dollars each. Landon Clay, the mutual fund magnate putting up the money, says “Unfortunately, the established revealed religions no longer provide the answers that are satisfactory, and that translates into a need for certainty and truth.” Certainty is available in mathematics, to the distress of those unhappy with such regressive notions. So is beauty, which may perhaps encourage art lovers who are out of sympathy with the offerings of modern curators. Casti actually opens his book with Aristotle’s thought that “those who assert that the mathematical sciences say nothing of the beautiful are in error. The chief forms of beauty are order, commensurability, and precision.” A production with any one of those qualities may not earn a sale to a leading museum of contemporary art, but, Casti shows, a piece of mathematics with all of them can still achieve success in the profession farthest removed from the cultural degradation of the present age: mathematics.

Contrary to the impression given by elementary mathematics that the discipline deals mainly with simple problems, much of recent mathematics is concerned with how to cope with complexity. Spheres and cubes are simple, but weather prediction is not. Even shapes are not. Shortly after Hales announced his proof of the Kepler Conjecture, he had a call from the Ann Arbor farmers’ market. “We need you down here right away,” they said. “We can stack the oranges, but we’re having trouble with the artichokes.”