# Mathematical Curiosities. A Treasure Trove of Unexpected Entertainments

A. Posamentier has has devoted a life to mathematics education and he has written and co-authored many books on the topic at a rate of almost one book per year. When it comes to motivating students for mathematics, one arrives easily at topics that can be classified as popular or recreational mathematics. An early title like *Math Wonders to Inspire Teachers and Students* (2003) makes clear what I mean. Later he has with his co-authors discussed more coherent subjects like Pi, the Fibonacci numbers, the Pythagorean theorem, the golden ratio. Last year's book with the present co-author I. Lehmann was *Magnificent Mistakes in Mathematics* (2013).

The present book continues the ideas of *Math Charmers: Tantalizing Tidbits for the Mind* (2003) and *Mathematical Amazements and Surprises: Fascinating Figures and Noteworthy Numbers* (2009). Posamentier and Lehmann have collaborated on several previous book projects and form a well-oiled tandem in producing collections like this one. In producing several of the kind of books I just mentioned, I can imagine that as a math lover, you keep a notebook ready at all moments and take a note of whatever mathematical idiosyncratic tidbit one happens to come across. Moreover with the given record and the type of books that the authors have published, I would assume that new ideas and pointers were submitted by enthusiastic readers to feed the notebooks. Their *Magnificent Mistakes* gives a partial selection of —what I imagine to be— their notebooks. That book was focussing on what can go wrong in computations or when reasoning guided by intuition becomes illogic. Even great historical mathematicians have made mistakes. This book is another selection form this virtually inexhaustible set of mathematical curiosities. So what are these curiosities? They can be mathematical puzzles, or problems that have a counter-intuitive solution, or strange regularities in numbers, that could easily inspire numerologists, etc. In short: anything that amazes a math lover, that makes you raise your eyebrows, or flashes an exciting aha experience. And Posamentier and Lehmann being convinced math lovers let themselves easily be amazed, maybe sometimes more than the reader would admit. Probably math-phobics might have a less exciting “so-what” or “who-cares” experience.

How did the authors bring some order in what must be a chaos of interesting ideas to put into this collection? They arranged them in five chapters. A first one is about arithmetical curiosities. Of course there are many curious numbers that are particular for various reasons. In fact any number is particular, and if it were not, it would be a curious number because it is an exception to the rule. A reason can always be found up to the absurd like 65 is particular because it is the only number whose first digit is 6 and its second is 5. Numerologists especially are very apt in finding hidden messages behind practicaly any number. Of course here the authors find better reasons to give a number the label of being curious. These are mostly special patterns or arrangements that appear doing computation. It is considered curious to note the arrangement of all 9 digits in the equality 192 + 384 = 576 which involves 3 multiples of 192. Or, if in 16/64 you cancel the 6 in numerator and denominator you get 1/4, the correct answer. But there are also other somewhat more serious number theoretic examples like perfect numbers, lucky numbers, happy numbers, number sequences,... or ways to compute prime numbers or other computational tricks like Babylonian or Russian peasant multiplication. Somewhat playful it is to compute the numbers from 0 to 100 using all possible algebraic operations on only one digit (like 5 = (4*4+4)/4 if the unique digit is 4). Once a curiosity is observed in some example, the authors transfer to the reader the mathematician's attitude to ask if this particular pattern is unique, or are there infinitely many solutions, or how can this be generalized etc.

The second chapter is a cabinet of geometric curiosities. There are some old Japanese geometric problems called *sangakus*, there is Kepler's sphere packing problem and its two-dimensional analog, or squaring the square (the problem is to tile a square with squares all of which have integer side lengths but with the least possible repetition of identical squares) and several other problems with circles, quadrilaterals, and triangles.

While the previous chapter is relatively short, the third one is rather extensive. It is a classical collection of mathematical puzzles of all sorts. Many such collections are available already in the literature. Its title is *Curious problems with curious solutions*. The problems are not *that* curious, but the crux most often lies in the less straightforward way in which they can be solved. For example, given the the sum and the product of two numbers, find the sum of their reciprocals. The straightforward way to solve this is solve a quadratic equation to give these numbers, compute their reciprocals and add. However, the sum of the reciprocals is the ratio of their sum over their product so that the result is immediate. This is just one example but there are 81 (mostly more complicated) such problems. Their solutions are also given but separated from the problem formulation to stimulate the reader and prevent that she should be tempted to peak at the solution before she has tried to solve the problem on her own.

Chapter 4 is again short since it treats a particular subject: How can the arithmetic, geometric and harmonic mean be retrieved using geometrical arguments, i.e., using a right-angled triangle, or rectangles etc. This is a bit more of mathematics and geometry, and a bit less carefree hopping in the mathematical playground.

The last chapter is devoted to fractions, in particular unit fractions play a central role. The harmonic triangle (an entry in this table is a unit fraction that is the sum of its two unit fractional children) is linked to the Pascal triangle (each entry is and integer, that is the sum of its two integer parents) and other relations illustrating that “there is more in fractions than meets the eye” as the authors conclude this chapter.

All the mathematics used is very basic and can be appreciated by anybody. The hope is that this will increase the love for mathematics. As much as I hope a book like this will help to pump up its popularity, I have some doubts. These books mainly attract the readers who love solving puzzles, but these are usually the ones who are mathematically oriented already. It might help to make math lessons a bit more interesting and challenging, but many of the problems and examples are somewhat “off the beaten path” and so might not always fit in a course where certain theorems and other chunks of less amusing theory have to be assimilated. Transferring an attitude of mathematical curiosity is certainly laudable. and the problem solving techniques of the third chapter may be helpful in teaching mathematics although some problems and techniques are too “curious” to have a wider applicability. However, it will be true fun reading for anybody with a mathematical mind. You do not need an advanced mathematical education at all. If you did study mathematics you will enjoy it too, but you might find some items a bit too low level, but there are certainly others that are new and/or to you too.

**Submitted by Adhemar Bultheel |

**25 / Aug / 2014