This is another book written to promote and popularize mathematics to laymen and unbelievers. In most cases, the author of such books is a mathematician who has to be careful not to introduce technical terms without explanation. This time the author is not a mathematician, which helps of course to guarantee that the book really stays at a level accessible to anyone. As an example, at some point it is even explained what a difference is between a conjecture and a theorem. Owen O'Shea is employed by the Irish Department of Defence and he has published a similar book before The Magic Numbers of the Professor (2007), and published several papers on recreational mathematics. So he has had some training.
The topics discussed are the classics: prime numbers, Fibonacci and Lucas sequences and Pascal's triangle, Pythagorean triples, triangular numbers, magic squares, the Monty Hall problem, and transcendental numbers: φ, π, e, √2, and the complex √-1. Most of these are also discussed in several other books. The lovers of this kind of books will recognize large parts, but there are some exceptions.
Each of these topics is elaborated in a separate chapter. The keyword throughout the book is "patterns". The strategy is always the same. First some elements are written by the author and it always ends with comments and additions by Dr. Cong, a numerologist and obviously a friend of the author. O'Shea gives his biography at the end of chapter 2: a mathematical child prodigy originating from China, who immigrated to the USA. A dramatic ski accident prevented that he became a professional mathematician. Instead he participated in a traveling carnival where he took on the nickname Dr. Cong. The emails of Dr. Cong that are sometimes included use a style and assumes a level of readership that is suspiciously similar to what the O'Shea wrote, which blatantly confirms that he is just O'Shea's alter ego. Also in The Magic Numbers of the Professor the `Professor' is a fictitious character.
The comments of Dr. Cong usually start by saying that the text is interesting but... and then he adds some extras to the topic discussed. His part is sometimes as long as the 'original text' that he is commenting on. However, he also adds (in my view very un-mathematical) numerological curiosities. For example the lo shu is an ancient Chinese 3 by 3 magic square that was exposed on the back of a turtle with the rows 4 9 2, 3 5 7, and 8 1 6. One of Cong's comments is that it refers to 666, the number of the beast because $4^3+9^2+2^1+8^3+1^2+6^1=666$. Or he comments in the third chapter that the first two letters of Pythagoras are the 16th and the 25th in the alphabet while the smallest Pythagorean triple is (9,16,25). A genuine mathematician's reaction to such statements would probably be: So what? And there are several other instances where dates, hours and other numbers can be combined to give so-called curious coincidences. In fact, there is a whole chapter on such "coincidences" which in my opinion diminishes the value of the book. These are indeed patterns, and it may attract extra readers who see tarot-like proofs in almost anything of whatever ethereal truth that is bestowed upon us by fate. It may (and should) shy away any readership of (potential) mathematicians. Of course it does make sense to explain that some coincidences are not as curious as one might think. For example the probability that two people in a group celebrate their birthday on the same day of the year is surprisingly high. That can be explained by simple statistics, but number fetishism is not mathematics. With this book, it is O'Shea's intention to make readers enthusiast for mathematics and take up interest in studying more of it. Solving a puzzle or finding out how a certain trick works may indeed be helpful. And there is indeed something to say about the recognition of patterns, but only if there is some rationality behind it that has to be discovered. But this kind of mysticism is a bad idea, or at least gives a very wrong impression of what mathematics is about. It is very easy to make such things up. For example, the figure in which Einstein's equation $E^2=(mc^2)^2+(pc)^2$ is represented on a Pythagoras triangle happens to be numbered 3.14, and moreover $3^1+4=7$, and 3.14 approximates π with 2 decimals in the fractional part, put 7 and 2 together and lo and behold, the figure appears on page 72. What a coincidence! But it is roaring nonsense.
There are of course also quite nice things to say about this book. First of all there is its very elementary approach, but it is still discussing many mathematical objects and ideas. Sometimes they are only mentioned or briefly touched upon. Not really analytic proofs of course, but sometimes strong suggestions and indications are given for limiting value. Similarly we meet also Einstein's relativity theory as I mentioned above, but also several different geometric proofs of the Pythagoras theorem, continued fractions, Platonism, complex numbers, $i^i$, Schrödinger's equation, Stirling's asymptotic formula for the factorial, and many others.
The author classifies his book under recreational mathematics. As he writes in the introduction, its purpose is to be entertaining and at the same time educational. He is obviously an admirer of Martin Gardner. But O'Shea's attitude towards mathematics is not as bad as I might have suggested above. To illustrate his vision on mathematics and how he sees his contributions in this book, I can quote what he writes on page 121. After it has been suggested (by numerical evidence) that the n-th root of the n-th Fibonacci number tends to φ (the golden ratio) as n tends to infinity, O'Shea writes "Of course there are those who ask what these curiosities tell us about our world. My answer to these questions is: They perhaps tell us nothing! Mathematics in itself does not explain our universe. Yes mathematics can be used in physics to explain how some parts of the universe operate. That is truly marvelous. But that is not why mathematics exists. Mathematics exists in its own right. It may well be the only reality. To find explanations on how the world works, I suggest one should study physics."
There are a few of problems or puzzles to solve (but not many) which get solutions at the end of the chapter, and for the hungry reader there is a list of references to read more. Thus, if you are interested in mathematical issues and puzzles of the mathematical type, you will certainly enjoy this book even if you have only a minimal background. Just be aware that numerology is as alien to mathematics as penguins are to the Amazon jungle.