The Joy of Mathematics
The authors directly address the secondary school student pointing them to mathematical issues that are not covered by traditional curricula. They are of course addressing students in the USA, but most of what they mention applies to the European system as well. I doubt it that most of these young adults will spontaneously read this book for fun, but there are always exceptions of course. Clearly, through these students, the authors are indirectly reaching the teachers, or it may well be the other way around.
The subtitle of the book: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class catch the spirit. What are all these tricks, techniques, and theorems which are not usually covered in a regular curriculum because of a lack of time? The authors have organised them in five chapters collecting many of them around a central theme. The first chapter is called Arithmetic Novelties. I hesitate to call these "novelties", unless they are novelties for the student who may read about them here for he first time. The "novelties" are classic arithmetic tools but that may have been forgotten because many computations are performed on computing machines nowadays and not so much in the heads of students anymore. Examples are shortcuts for divisibility checks, formulas to sum numbers or squares of numbers, the Euclidean algorithm, and fun things to know about numbers like palindromic, triangular or square numbers, perfect numbers and the likes, and more material of that style.
The second chapter collects some algebraic items. Here are some classics like the irrationality of the square root of 2, why a division by zero allows to prove anything true or false, and there are again useful computational methods: the bisection method to find a zero, the Horner scheme for polynomial evaluation, and problems like solving Diophantine equations, generating Pythagorean triples, Descartes's sign rule for zeros of polynomials, and more.
The geometry topics of chapter 3 take more pages, but that is mainly because these require many graphical illustrations. As you might expect, we find here several less conventional proofs of the Pythagorean theorem and several of its possible generalisations. Also many theorems involve circles (not surprising since the authors published a year earlier in 2016 The Circle. A Mathematical Exploration Beyond the Line, a book completely devoted to such circle theorems). But there are many other properties as well that involve triangles, spirals, polygons, Platonic solids and star polyhedra, and much more.
The chapter on probability is relatively short. Here the surprise effect of unexpected results are a central theme. Benford's law, coinciding birthdays, the Monty Hall problem and the related paradox of Bertrand's box, the false positive paradox, and the poker wild-card paradox. Other topics are surprising properties of Pascal's triangle and random walks.
The last chapter is a collection of miscellaneous problems. About the origin of some of the familiar mathematical symbols, compound interest and the rule of 72 to double your investment, the Goldbach conjecture, countability and the different levels of infinity, properties and constructions of the parabola, the speed of a bicycle as a function of the sprocket wheel used, and several others.
Anyone who is a bit familiar with the literature on popular and recreational mathematics will find that most items collected in this book are not really novelties, and as a gem, they are not really neglected, but they certainly are rarely taught in math class. However, if you know some teenager who loves mathematics, then this will be a fantastic gift. All the content is up to the level of her mathematics and it are marvels and gems, which are most probably novelties to her. The good thing is that everything is not just raising wonder and surprise, but it is explained why it works and proved when appropriate. It is not a regular textbook tough with formal theorems, proofs and exercises. It is kept at an entertaining level. If, as a teacher, you have some spare time within the strict framework of the curriculum, you can use the book as an inspiration for examples that are stimulating the interest of your pupils.