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How Round Is a Cube?

James Tanton
Publication Date: 
Number of Pages: 
MSRI Mathematical Circles Library
[Reviewed by
John Ross
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How Round is a Cube? is a charming collection of mathematical “ponderings,” neatly organized into 34 “essays.” The text was published as part of the MSRI Mathematical Circles Library series, and the text certainly sparks a creative, playful approach to mathematics that is appropriate for a math circle (or math teachers circle). However, the author’s motivation is not purely to prompt such circles – and he himself admits that the collection and cataloging of these problems stems primarily from an interest in (and a joy of) mathematical gems with quirky twists. The result is a thought-provoking book that will pique the interest of math circle participants and solo math enthusiasts.
The text does not subscribe to a single pool of mathematical thought. Instead, the author draws from many wells in these essays, finding inspiration from algebra, combinatorics, number theory, geometry, and statistics. A detailed list of topics appears at the front of the text, so a reader interested in pursuing a problem using geometric probability (for example) will know which essay to turn to. At the same time, the wandering reader can open the book to an essay at random and be exposed to a new and unexpected branch of mathematics. 
The organization of each essay is impressive and should be noted. Each begins with a problem (or a handful of problems) that is/are simply stated and easily approachable. Since each essay is self-contained and requires very little background knowledge, these problems all have a low barrier to entry: you’ll be immediately hooked, and you’ll pause before turning the page so you can see what conclusions you are able to draw on your own. If and when you do turn the page, you’ll be treated to an introduction of a concept in mathematics (the Chinese Remainder Theorem, say, or the Leibniz’ Harmonic Triangle) that can be used in a clever way to solve the previously stated problem. Each chapter concludes with a “research corner” in which more advanced questions are asked and no answers are given – it’s a launchpad to further exploration.
As stated earlier, this text has a clear use as a book that offers prompts and solutions for math circles. The initial problem that kicks off each essay could also be used to kick off a circle session. And yet, it would be a mistake to consider this simply a “book of problems for math circles.” The clever mathematical connections discussed in this text make it a worthy read for any math enthusiast with broad interests and a love of clever mathematics.


John Ross ( is an Assistant Professor of Mathematics at Southwestern University. His research interests lie in the field of Geometric Analysis.