It should be easy for a mathematician to write a book on recreational mathematics, right? Ha! You can look through my reviews for some godawful examples of this genre.
It is not enough to know mathematics. The writer has to write well and to have an understanding of the mindset of those of us who are strictly amateurs. The material has to have intrinsic interest and the explanations have to be clear and well motivated but not condescending. The problems should be challenging but not impossible and they should serve to gain insight into the material. And they should come with answers!
All of this Ball accomplishes. The quality of the writing reminds me of Martin Gardner. There is one thing that I need to emphasize. This book contains proofs, because proofs are the heart and soul of mathematics. The proofs are exceptionally clear, but if mathematical proofs are not your thing then go elsewhere. If you are not intimidated by proofs and have a knowledge of high school matematics up through calculus then you are in for a real treat.
I particularly liked the chapter on continued fractions. This is the first time that I have seen a treatment of the matrix approach. I do have one very small quibble. The matrix approach allows for a very simple derivation of the formula for convergents, which I was able to discover for myself but which is not presented in the book. All of the other proofs I have seen for the formula do not provide a derivation, but instead do a proof by induction. I hope that future editions of the book include the derivation of the convegents formula.
