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**The mathematics of various entertaining subjects. Volume 3. The magic of mathematics. With a foreword by Manjul Bhargava.**
*(English)*
Zbl 1446.00004

Princeton, NJ: Princeton University Press; New York, NY: National Museum of Mathematics (ISBN 978-0-691-18257-5/hbk; 978-0-691-18258-2/pbk; 978-0-691-19441-7/ebook). xxi, 325 p. (2019).

The Mathematics of Various Entertaining Subjects (MOVES) is a conference that happens every two years, the next one being in 2021. The present volume with the same title and subtitled ‘The Magic of Mathematics’ is the proceedings of the 2017 MOVES conference. So, this volume is basically a collection of research articles about various aspects of recreational (but at the same time serious) mathematics.

A crisp review of the book can be summed up in one sentence: This is a fantastic (and entertaining) book on various aspects of recreational mathematics which are also at the forefront of research level mathematics, with topics ranging from puzzles, games, algebra, number theory, topology, geometry and combinatorics. But it would not do justice to this excellent collection. In fact, any short review as this one would not be able to do justice to the eclectic mix of topics covered by experts in this collection.

Recreational mathematics and other ‘serious’ type of mathematics has had its demarkations shrink in recent years. In fact, some branches of mathematics like probability came into existence as a rigorous mathematical discipline via escapades in recreational mathematics. This collection will further shrink this gap between recreational and serious mathematics, a cursory glance at the list of authors is sufficient to cement this statement. Masters of the subject and young mathematicians alike have come together and given us a unique (well, other volumes of the series also exist as this being the 3rd volume might suggest) insight into this world of recreational research mathematics.

Having said all of this, it should not come as a surprise that this book is not for the novice or for the faint hearted. One needs to have a good grasp of algebra (group theory and linear algebra), probability theory, computational complexity and graph theory to really appreciate some of the articles in the collection. For others still, a short primer is helpfully included, but at a very fast pace. However, the first six chapters on puzzles can be easily enjoyed by anyone interested in mathematics or puzzles.

The review will not aim to cover each and every article in the collection, it would be impossible to really do this by one single person in such a short review. What we do instead is to give a one line summary of some of the articles which the reviewer found particularly intriguing. As in all such cases, this selection is heavily influenced by personal taste and choice.

The collection itself is subdivided into four parts: puzzles, games, algebra and number theory, and geometry and topology with various aspects of discrete mathematics coming into play regularly.

In the first part on puzzles, the chapter by Steven T. Dougherty and Yusra Naqvi gives a very nice and short introduction to coding theory via two puzzles, while the chapter by Tanya Khovanova studies various aspects of coin weighing problems and presents several scenarios going from easy to hard which the reviewer found particularly interesting.

In the second part on games, David Molnar explains the Burnside’s Lemma using applications to counting the number of possible tiles in several different games, while Michael P. Alloca, Steven T. Dougherty and Jennifer F. Vasquez study a game which they call ‘Japanese ladders’ using permutations, braid groups and graph theory. It appears to the reviewer that these two chapters could feature as quite an entertaining and motivated introduction to several undergraduate topics in mathematics.

In the third part on algebra and number theory, Persi DIaconis and Ron Graham discusses the life of Charles Sanders Pierce using magic tricks. This chapter is particularly instructional if read with a deck of cards and doing all the things mentioned in the chapter. Another chapter in this part by Max A. Alekseyev proves an old conjecture of Ron Graham concerning certain questions of partitioning integers into squares whose reciprocal sum up to \(1\).

In the final part, Yossi Elran and Ann Schwartz gives a quick primer on knot theory and studies knots and flexa-bands, while David M. McClendon and Jonathon Wilson studies the combinatorics of Legos. All the parts decrease in terms of number of chapters, starting with six in the first part and ending with three in the final part.

As can be seen from the last few paragraphs, the mix of topics is diverse and we have not touched upon many of the chapters in this collection. The best thing to do would be to just pick up the book and browse randomly at anything that catches one’s fancy. Given the choices, the reviewer is sure that every mathematician will find something in the book that will interest her.

A crisp review of the book can be summed up in one sentence: This is a fantastic (and entertaining) book on various aspects of recreational mathematics which are also at the forefront of research level mathematics, with topics ranging from puzzles, games, algebra, number theory, topology, geometry and combinatorics. But it would not do justice to this excellent collection. In fact, any short review as this one would not be able to do justice to the eclectic mix of topics covered by experts in this collection.

Recreational mathematics and other ‘serious’ type of mathematics has had its demarkations shrink in recent years. In fact, some branches of mathematics like probability came into existence as a rigorous mathematical discipline via escapades in recreational mathematics. This collection will further shrink this gap between recreational and serious mathematics, a cursory glance at the list of authors is sufficient to cement this statement. Masters of the subject and young mathematicians alike have come together and given us a unique (well, other volumes of the series also exist as this being the 3rd volume might suggest) insight into this world of recreational research mathematics.

Having said all of this, it should not come as a surprise that this book is not for the novice or for the faint hearted. One needs to have a good grasp of algebra (group theory and linear algebra), probability theory, computational complexity and graph theory to really appreciate some of the articles in the collection. For others still, a short primer is helpfully included, but at a very fast pace. However, the first six chapters on puzzles can be easily enjoyed by anyone interested in mathematics or puzzles.

The review will not aim to cover each and every article in the collection, it would be impossible to really do this by one single person in such a short review. What we do instead is to give a one line summary of some of the articles which the reviewer found particularly intriguing. As in all such cases, this selection is heavily influenced by personal taste and choice.

The collection itself is subdivided into four parts: puzzles, games, algebra and number theory, and geometry and topology with various aspects of discrete mathematics coming into play regularly.

In the first part on puzzles, the chapter by Steven T. Dougherty and Yusra Naqvi gives a very nice and short introduction to coding theory via two puzzles, while the chapter by Tanya Khovanova studies various aspects of coin weighing problems and presents several scenarios going from easy to hard which the reviewer found particularly interesting.

In the second part on games, David Molnar explains the Burnside’s Lemma using applications to counting the number of possible tiles in several different games, while Michael P. Alloca, Steven T. Dougherty and Jennifer F. Vasquez study a game which they call ‘Japanese ladders’ using permutations, braid groups and graph theory. It appears to the reviewer that these two chapters could feature as quite an entertaining and motivated introduction to several undergraduate topics in mathematics.

In the third part on algebra and number theory, Persi DIaconis and Ron Graham discusses the life of Charles Sanders Pierce using magic tricks. This chapter is particularly instructional if read with a deck of cards and doing all the things mentioned in the chapter. Another chapter in this part by Max A. Alekseyev proves an old conjecture of Ron Graham concerning certain questions of partitioning integers into squares whose reciprocal sum up to \(1\).

In the final part, Yossi Elran and Ann Schwartz gives a quick primer on knot theory and studies knots and flexa-bands, while David M. McClendon and Jonathon Wilson studies the combinatorics of Legos. All the parts decrease in terms of number of chapters, starting with six in the first part and ending with three in the final part.

As can be seen from the last few paragraphs, the mix of topics is diverse and we have not touched upon many of the chapters in this collection. The best thing to do would be to just pick up the book and browse randomly at anything that catches one’s fancy. Given the choices, the reviewer is sure that every mathematician will find something in the book that will interest her.

Reviewer: Manjil Pratim Saikia (Cardiff)

### MSC:

00A08 | Recreational mathematics |

00A09 | Popularization of mathematics |

00B25 | Proceedings of conferences of miscellaneous specific interest |

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\textit{J. Beineke} (ed.) and \textit{J. Rosenhouse} (ed.), The mathematics of various entertaining subjects. Volume 3. The magic of mathematics. With a foreword by Manjul Bhargava. Princeton, NJ: Princeton University Press; New York, NY: National Museum of Mathematics (2019; Zbl 1446.00004)