Review
“The Author considers these problems to be the type that two mathematical friends would pose to each other and discuss over a cup of coffee in a lounge. I agree with that premise, they are not too hard and there is a proof that is relatively easy to discover and even easier to understand. These problems satisfy all of the requirements for a good problem…” Journal of Recreational Mathematics
Can a Christian escape from a lion? How quickly can a rumor spread? Can you fool an airline into accepting oversize baggage? Recreational mathematics is full of frivolous questions where the mathematician’s art can be brought to bear. But play often has a purpose. In mathematics, it can sharpen skills, pro [Read More...]
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April 16th, 2010 at 8:28 pm
Back in the days when I was a mathematician I remember taking part in many coffee room discussions with problems similar to the ones found in this book. Sometimes problems like these arise from a detail of a concrete problem that a mathematician is working on, and other times the coffee room problem comes first and turns into something much bigger and deeper than it first appeared.
A good coffee room problem is one whose statement can be understood by all present (from the lowliest honours student upwards) so that everyone can take part in thinking about the solution without needing specialised knowledge. The problem should also be hard enough to keep everyone occupied for a while and should be more profound than it looks.
This book is a collection of such problems. The book is divided into three sections: problems, hints and solutions. Even if you can’t solve a given problem, just the act of working on it for a few minutes, hours or days (depending on your stamina) before checking the answer still gives you a lot of insight into the problem and a great deal of enjoyment. The solutions often go deeper than just the original problem. For example, the first problem of the book about a lion chasing a christian in a circular arena turns out to be quite a famous problem that has been generalised to higher dimensions, different types of arenas and greater numbers of lions!
So you won’t be forced to give up on all of them, some of the problems are much easier than others, but all will get you thinking. Although many are even accessible to the advanced high school student, a fair bit more sophistication is often required, particularly to fully appreciate the solutions. I believe Bollobas intended his primary audience to be the type of people you might meet at a university mathematics department coffee room, but it is also perfect for lapsed mathematicians like me who still remember enough mathematics to tackle this book. In this sense I found it a real nostalgia trip.
April 16th, 2010 at 10:21 pm
The 157 problems in this book are certainly challenging and it is clear to see how they can be characterized as problems one mathematician would pose to another over coffee. Nontrivial with unusual points of statement and solution, most have proofs that are short and direct. The coverage is broad; many areas of mathematics are represented in some form.
The first section of the book is the set of problems, followed by a list of very brief hints. Given that it takes 35 pages to list the problems and the hints only take up nine pages, the hints are often of dubious value. The last section is a set of complete solutions to the problems, if you cannot solve them; they are an opportunity to appreciate how well done the problems are and the educational value of the proofs.
“Here’s a cool problem” is a game that mathematicians like to play with each other and those found in this book are definitely in that category.