Mathematical Curiosities: A Treasure Trove of Unexpected Entertainments

Mathematical Curiosities by Alfred S. Posamentier & Ingmar Lehmann


I promise not to tell a lie. I found this book a bit frustrating to read. The frustration stems from the fact that half of the time I was mesmerized by the mathematical curiosities promised by the title while the other half of the time I found them rather tedious and tiresome. This may be because my affection for beautiful mathematics is perhaps not as high as I myself expected, but the problems I found with the book also stem from its basic structure. Therefore, this review’s structure will be a bit binal in nature.

(Get it? Bit… binal…binary… Sorry, I’m just finding myself unendurably funny these days).

The book is divided into five chapters, each one devoted to its own branch of mathematical curiosities, kind of.  These are entitled “Arithmetic Curiosities”, “Geometric Curiosities”, “Curious Problems with Curious Solutions”, “Mean Curiosities”, and “An Unusual World of Fractions”. The chapter on problems and solutions was a most dissatisfying travail. The chapter is set up as a series of 90 mathematical problems with very contrasting levels of curiosities. Some are kind of obvious, some are interesting enough while yet others just… aren’t (in my opinion). My disgruntlement with the chapter is that I wanted to know the answer once I tried solving the puzzle in my head. The authors, however, don’t trust the reader enough to show the solution right after the puzzle because one might have an inkling to sneak a peek. This I found insufferable. Once I thought I had the solution (or didn’t want to spend more time of a particular puzzle, flipping back and forth between the puzzles and the solutions is just tiresome and defeats the point anyway of separating the puzzles from the solutions.

Besides this major disagreement between myself and the authors I found their gratuitous self-citations a bit bothersome. „I you want to know more about some interesting stuff, we’ve already written about so buy our other books too. I mean. I’ll admit to self-citing my previously published papers for the citation stats. But in book form, I think it’s kind of lame.

Despite all my aforementioned grievances, I want to give credit where credit is due. The good parts of the book (which accounts to roughly 70%) are electrifyingly interesting. The authors show verve and passion in their presentation of especially the arithmetic curiosities, which are just breathtaking. Strange, weird, hypnotizing. Arithmetic can take the form of artistry and mind boggling magic. When the book is good, it’s magnificent. When it’s not, it’s frustrating.

There is a wealth of number patterns that are explored that left me amusingly bewildered and bewilderingly amused.

number_pattern_10number_pattern_9Courtesy of:

The concept of the Japanese Sangaku tablets are examined and many examples dissected. Sangakus are geometric problems that are presented as geometric puzzles, comprising multicolored triangles, polygons, circles, circular arcs, etc. There is an inscribed geometric beauty about them and a deep mathematical intrigue. When Sudoku falls out of favor. Sangakus could replace them.


Left panel. Ancient Sangaku tablet. Right panel. A modernized Sangaku puzzle. 

The harmonic triangle also makes an appearance. The harmonic triangle is very similar to Pascal’s triangle only it makes use of fractions rather than whole numbers. Each member of the triangular arrangement of unit fractions will be such that the sum of the two fractions below it – the one to the right and the one to the left of the member – will be equal to that fraction. The similarities between the two triangles are quite peculiar and intruiging in their own way as is explored in further detail here.

581098eb5e9213bf6c66e932ed218e08The Harmonic Triangle

In summary, the peculiar numbers, sequences of arithmetic operations, and even some of the curious problems, do tilt my opinion so that I would recommend it to anyone. But I will expect different people to have different reactions to the book. There are enough interesting numbers for anyone I suppose.


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