Cheeky problems and dotty solutions

Keith Ball writes: "I gave a number of 'popular' lectures in schools, or to school pupils visiting University College London. This book is based on those lectures. Although [it] is meant to be recreational, my aim has been not only to entertain, but also to convey some of the ideas that lie just beyond what is normally taught in the last few years of school." Given that the audiences were members of maths clubs or those who visited UCL, he could start with a presumption of elementary calculus and the algebra of two-by-two matrices. For such readers, this book should be enjoyable.

Even for those without these prerequisites, the first three chapters are accessible and entertaining. Chapter one, "Shannon's free lunch", starts by explaining the ISBN codes used to identify books, then goes on from there to error-correcting codes.

Chapter two, "Counting dots", is about the beautiful but little-known Pick's theorem, which states you can calculate the area of a very complicated polygon by just counting the lattice points inside and on the boundary.

The next chapter, "Fermat's little theorem and infinite decimals", is about a puzzle that has always intrigued me. If you work out the decimal expansion of a fraction such as 1/7, you get a repeating sequence of six digits, 142857. What explains this particular sequence? And why does it have six digits? Is it because 6 = 7 - 1? The repeating sequence for 1/13 is 076923. It also has six digits. Is that because 6 = (13 - 1)/2? Notice that 142 + 857 = 076 + 923 = 999. Why?

The later chapters explain other slightly offbeat topics: space-filling curves, the bell curve, Stirling's formula, continued fractions and approximation of irrationals. Excellent for the bright, ambitious school-leaver.

Throughout, Ball maintains an accessible and light-hearted style. For example, after explaining that a number must be irrational if it can be approximated infinitely closely by rational numbers, he writes: "To my mind, this is one of the cheekiest arguments in mathematics."

Reuben Hersh is emeritus professor of mathematics, University of New Mexico, US.