"When I was fourteen years old, I started a notebook. A _math_ notebook." Ian Stewart starts his most recent book this way, and then apologizes for being such a geek. He has written lots of books about serious mathematics, and his new one is serious, too, but it is full of serious fun. _Professor Stewart's Cabinet of Mathematical Curiosities_ (Basic Books) comes from that notebook, and the subsequent notebooks he had to get because more curiosities kept crowding in. He didn't put his school math in the notebooks; he put in all the interesting math that he wasn't taught at school. So in these pages are about two hundred short chapters or essays on what is usually called "recreational math". It's not mathematics you can be tested on, so it's fun. A lot of it does not have to do with numbers; mathematicians may forever be associated with numbers and counting, but it is the logic and the study of patterns that occupies higher math, and a lot of that higher math can be brought down to earth for entertainment purposes, as Stewart has done here repeatedly. For those who like recreational math books, there will be much that is familiar, like the problem of crossing all the bridges of Konigsberg exactly once, or that of the farmer who has to cross the river with a wolf, a goat and a cabbage, but has room in his boat for only two at a time, and none must get eaten by the others en route. If those don't ring a bell, this is a splendid book to start you on wondering about some entertaining mathematical ideas. If you know the old ones, Stewart has included lots of new puzzles, as well as small biographies of quirky mathematicians through history, and little essays on non-puzzle material like fractals or Gödel's proof. He has also, at the back of the book, included the answers, in a section labeled, "Professor Stewart's Cunning Crib Sheet: Wherein the discerning or desperate reader may locate answers to those questions that are currently known to possess them... with occasional supplementary facts for their further edification."
There are rings on the coat of arms of the Borromeo family, three rings that you cannot pull apart but none of which is linked to another. There is a section on famous mathematicians who aren't famous for being mathematicians. Sure, you knew Lewis Carroll, famous for the _Alice_ books, was a mathematician / logician, but did you know Art Garfunkel got his master's in math, and only stopped work on his PhD so he could pursue his singing career? Bram Stoker, author of _Dracula_, had a mathematics degree. Leon Trotsky had his mathematical career ended by exile to Siberia. There is a section on Fermat's famous Last Theorem and how it was proved fifteen years ago by complicated modern methods. Fermat himself could not have used such methods in the proof he said he had, but he did not write it down because he didn't have enough space in the margin in which he was writing notes. Stewart says that there might be a simpler proof, and while he repeatedly encourages readers to branch out on their own from these problems, he warns them about coming up with proofs for this one, and he also hints at the frustrations of being a public mathematician: "If you think you've found it, _please don't send it to me_. I get too many attempted proofs as it is, and so far - well, just don't get me started, OK?" There is a section on dividing a cake fairly. It's easy with two people - one cuts the cake and the other gets to decide which piece to take. How do you extend this to three people? If you have a block of cheese in cube form, how can you cut it so that the cut face is hexagonal? Why in lists of numerical data, like the areas of each of the fifty states, are the numbers far more likely to start with 1 or 2 rather than 8 or 9? And how can this be true whether the numbers represent square miles, square kilometers, acres, or any other measurement? What shape of road would give a smooth ride to a bicycle with square wheels? A person born in 35 BC died after his birthday in 35 AD; how old was he? (Hint: those ancients could do math, but they didn't have the concept of 0.) What number, spelled out in Scrabble tiles, equals its Scrabble score? This delightful book is a real miscellany.
It also has one characteristic those older recreational math books didn't have: internet references. When discussing, for instance, John Horton Conway's fascinating complexity-from-simplicity game Life, Stewart can send the reader to an internet version "which is easy to use and will give hours of pleasure." Some of the references are merely to Wikipedia, but others are to specialty sites, including the extensive Wolfram Mathworld. This would be a wonderful book to give to any young person, especially one who claims not to like math. Stewart may not have a cure for such a condition, but his fine collection of amusements could demonstrate that such abhorrence is at least sometimes misdirected.