A dish of water and tea may win you $1 million

Would you like to win $1 million? Have you always been good at sums?

Actually, it will take rather more than being good at sums to crack any of the seven problems described in this book. Before the Clay Mathematics Institute announced the problems in May 2000, and attached a $1 million prize to each one, it had consulted some of the world's top mathematicians.

Keith Devlin's book aims to give some idea of what the seven problems are, and why mathematicians think they are so important.

He begins with the Riemann Hypothesis, which is a 140-year-old conjecture related to the distribution of the prime numbers. He then moves on to Yang-Mills theory (modern mathematical physics) and the P versus NP problem (computability). He conjectures that the last of these is likely to be the only one of the seven that might conceivably be accessible to an amateur mathematician, as opposed to a highly specialised expert.

Then he comes to the Navier-Stokes equations, which govern fluid motion.

Take, for example, a shallow dish and fill it with cold water to a depth of 1cm or so. Sprinkle tea leaves on top, to visualise the flow. Then hold a pencil upright, dip it in the water, and move it across the dish. If you do this at a moderate speed you will observe a strikingly regular pattern of vortices in the wake of the pencil, spinning in alternate directions. While mathematicians can now (just about) account for phenomena of this kind, fundamental questions about the governing equations remain, and some of the deepest questions are about when solutions exist, even, let alone when those solutions might correspond to what is observed.

After a look at such matters, Devlin moves on to the Poincare Conjecture (topology). Since the book was published, there has been much excitement about this problem, and a proof by the Russian mathematician Grigori Perelman is still being checked by other experts in the field. Devlin follows with the Birch and Swinnerton-Dyer Conjecture (number theory), and then finishes with the Hodge Conjecture (don't even ask).

Devlin is a highly successful and well-known populariser of mathematics, and in each chapter he reviews some suitable elementary parts of the subject before sketching the prize problem itself. Any reader interested mainly in those more "general" parts might be better off with one of Devlin's other books, such as The Language of Mathematics (1998), but if what you want is a slight glimpse (very slight, in some cases) of the seven millennium problems, then I find it difficult to imagine any other book doing a much better job.

Devlin is candid about the difficulties he faced when writing the book. In connection with the Birch and Swinnerton-Dyer Conjecture, he notes:

"Although I have been a professional mathematician for more than 30 years, number theory is not my area of expertise, and it took me considerable effort, spread over several weeks, aided by discussions with experts that I knew, before I understood the problem sufficiently to write this chapter."

For that kind of reason, some may question whether anyone should be attempting to write a book such as this. Can there really be any point in even writing down, say, the Navier-Stokes equations if one feels obliged to prepare the way with an account of elementary calculus?

But, for what it is worth, I am on Devlin's side here. I suspect that a lot of the difficulty with mathematics in schools and universities is caused by students getting so bogged down with minutiae that they lose track of where they are really going. "Don't run before you can walk" is a good dictum, but so is "If you have no idea of where you're going, don't be too surprised if you never get there". In this respect, books such as The Millennium Problems can only help.

The $1 million prize money attached to each problem (see www.claymath.org ) has already done a service by helping to publicise mathematics as a living subject that is full of exciting challenges. And, contrary to common opinion, mathematicians are not from another planet; like everybody else, they are attracted by lots of money.

In the end, though, anyone who solves one of these problems will almost certainly be driven more by a love of mathematics for its own sake, including its remarkable applications, the elegance of some of the deductive arguments involved and the sheer wonder of some of the theorems, particularly when they reveal unexpected connections between different parts of the subject. This last aspect - surprising connections deep within mathematics - comes out particularly well in Devlin's ambitious book.

David Acheson is a fellow of Jesus College, Oxford.