Second edition (2018)

  • Bibliothèque Tangente n°76: Itération et récurrence, Éditions POLE, Paris, décembre 2021; p. 77., excerpt

(English translation). . . have collected interesting and important contributions into a book that is suitable both for the enthusiast of recreational mathematics and for the researcher who wants to get to know the field. It contains an introduction to all the necessary knowledge for a mathematical analysis of the problem, proofs of results, as well as exercises for consolidation and reflection on what has been read.

Tilen Marc, Obzornik za matematiko in fiziko, Nove knjige

La bible des brahmanes ?

Les tours de Hanoï constituent un objet de recherche extrêmement riche, bien plus qu'on ne peut l'imaginer la première fois qu'on s'amuse avec quelques anneaux et trois piquets. Quelques spécialistes de la question, Andreas Hinz, Sandi Klavžar, Uroš Milutinović et Ciril Petr, ont uni leurs efforts pour rédiger un ouvrage devenu incontournable sur le sujet : The tower of Hanoi – Myths and Maths (Birkhäuser, 2013). Le livre contient une présentation exhaustive des résultats connus sur le jeu et de nombreuses démonstrations détaillées. Beaucoup d'algorithmes et leurs preuves sont proposés, y compris pour les nombreuses variantes évoquées. Les illustrations sont riches et agréables et l'aspect culturel et historique est bien mis en valeur. Surtout, le livre se suffit à lui-même pour chaque détail, les questions ouvertes sont mentionnées et des exercices sont même proposés. Les auteurs arrivent à captiver aussi bien les étudiants, les passionnés que les professionnels. Une deuxième édition a vu le jour en 2018, notamment après les avancées de Thierry Bousch sur le cas des tours a quatre piquets.

Tangente n°76: Itération et récurrence

This is a second, enlarged, edition of a wonderful book about the game The Tower of Hanoi (TH); the book mentions several other more and less known puzzles, prominently the Chinese rings puzzle.

The book . . . is devoted to the mathematics and computer science related to the TH puzzle and to many of its variants.

There are many beautiful color illustrations in the book. It can be warmly recommended to any puzzle enthusiast, to any combinatorialist, and, I dare to say/write, to any mathematician.

Martin Klazar, MathSciNet, Mathematical Reviews

First edition (2013)

  • Cory Palmer in The Mathematics Enthusiast, December 2014, Vol. 11, No. 3 (p.753-754), excerpt

This book is a thoroughly comprehensive monograph on the theory behind the recreational game “The Tower of Hanoi,” ... [which] isn’t just a recreational problem, it is also a substantial area worthy of study, and this book does this area full justice. The book is an unusual, but very welcome, form of mathematical writing: recreational mathematics taken seriously and serious mathematics treated historically. It seamlessly links the history of this topic together with contemporary research in graph theory and computer science. The book itself is attractively presented with colour photographs and diagrams ... This book contains some serious mathematics, but despite that, a motivated undergraduate student could certainly read and understand it. Given that each chapter contains exercises, and despite the fact that hints and solutions are provided, it would make an excellent textbook for an eclectic but very enjoyable undergraduate course that crosses over a number of areas of pure mathematics. I haven’t enjoyed reading a “popular mathematics” book as much for quite some time, and I don’t hesitate to recommend this book to students, professional research mathematicians, teachers, and to readers of popular mathematics who enjoy more technical expository detail. I suspect other areas of mathematics could be treated in this way, but this book has set a very high standard for other authors to follow. To quote Ian Stewart in the Foreword, “the world of recreational mathematics has a brilliant new jewel in its crown”.

Chris Sangwin, Mathematical Intelligencer

strangely it fills an unexpected hole in the literature

Frank Berkshire,

The Tower of Hanoi book has all you ever wanted to know about the history and mathematics of the puzzle. However, the book goes deep into the Math, so it is not an easy read.

Gregory P. Ludwa,

Despite its somewhat playful title, this book is not for a casual reader. It really should be thought of as a mathematical textbook.

I think the book would be ideal for a topics course for graduate students and advanced undergraduates. [...] This book would also serve as a good base for thesis or dissertation topics for different levels of students of mathematics.

One particularly unique and appealing part of the book is its dedication to telling the history of the Tower of Hanoi puzzle and related puzzles. Along the way they dispel several persistent myths in the history of mathematics. Most textbooks either have no historical perspective or include just a few footnotes. I believe that the approach taken by the authors should serve as a model for other textbooks.

Overall the book is well-written and the authors make good decisions about how to present the material.

The combination of the historical background and essentially graduate-level mathematics makes this book unique and a treat to read. As such, I would recommend it to anyone with interest in mathematical puzzles and some background in upper-division mathematics.

Cory Palmer, The Mathematics Enthusiast

At first view this puzzle seems to be one of those “mathematical” puzzles without real difficulty and/or interest. But this is absolutely not the case! In this nice book, the authors show a large number of mathematical results coming from or associated with the Tower of Hanoi.

Each time I open the book I discover a renewed interest in the Tower of Hanoi. I am sure that this will be the case for all readers, who will share my enthusiasm and enjoy all of the chapters, not to mention the numerous exercices, the bibliography with 352 references and the 21 conjectures or open problems listed at the end of the book.

Jean-Paul Allouche, Newsletter of the European Mathematical Society

[T]he book demonstrates [that] the Tower of Hanoi has a very rich mathematical structure, and as soon as we tweak the parameters we surprisingly quickly find ourselves in the realm of open problems.

[T]he book could serve two main categories of readers (with possible overlap between the two of course). The first are those interested in the Tower of Hanoi and its variants, trying to better understand the game or to attempt some of the research questions left open. The second are those looking for a book about interesting mathematics, random mathematical “gems”, accessible theorems and proofs about an entertaining topic. For the first category of readers the book is indispensable.

For the interested layperson, the book is accessible and mostly self-contained, the topic is engaging and the enthusiasm of the authors is contagious. [...] Thus, I would warmly recommend the book for anyone interested, or as a gift for say, a mathematically-inclined high school student or undergraduate.

The historical asides and anecdotes are interesting and sometimes funny.

Overall, [...] the book fulfills its intended purpose, the authors give an authoritative and comprehensive overview of an interesting [...] topic, they strike a good balance between intuitive explanations and formal clarity. The book invites further thinking and research and both the historical and the mathematical parts (the latter making up the majority of the book) are enjoyable to read.

László Kozma, ACM SIGACT News

This book is written by four of the leading experts on the Tower of Hanoi. For this reason, it is destined to be an indispinsible resource for researchers interested in the Tower of Hanoi.

It is not leisurely reading. This is a hardcore mathematical text.

It [...] would make an excellent text for a special topics class or as a resource for a student doing a thesis/dissertation on the Tower of Hanoi. For this reason, I recommend it.

[L]ike with all good math books, be prepared to spend a bit of time working through examples/exercises and making notes in the margins.

Robert Beeler,

This book takes the reader on an enjoyable adventure into the Tower of Hanoi puzzle (TH) and various related puzzles and objects.

The style of presentation is entertaining, at times humorous, and very thorough.

As such, the book will be an enjoyable read for any recreational mathematician but, as the authors point out, could also be used as a text book for a second course into a particular area of discrete mathematics or as an introduction for research into the various unsolved conjectures and the list of possible extension topics given in the final chapter.

[The back matter] provide[s] the reader with as much help as anyone might wish for in a mathematical text.

Andrew Percy, Zentralblatt MATH

This puzzle, however recreational in nature, has very interesting mathematics underlying it. While it is simple to state, it takes a whole book to explore.

Each chapter offers absorbing exercises at the end, and hints and solutions to some of the exercises appear in an appendix. The authors support their ideas with numerous photos and color illustrations.

The style of writing makes the content comprehensible even to novices.

I strongly recommend it to students and researchers of mathematics and computer science, and also game enthusiasts.

S. V. Nagaraj, ACM Computing Reviews

Previous books typically did not even devote a section, let alone a chapter, to this problem, so the work will certainly become the standard reference on the topic.

The authors explain all the combinatorial concepts they use, so the book is completely accessible to an advanced undergraduate student. For this reason, some sections can be used for a reading course.

Recommended. Comprehensive mathematics collections, upper-division undergraduates through researchers/faculty.

Miklos Bona, Choice Reviews Online

The Tower of Hanoi is an example of a problem that is easy to state and understand, yet a thorough mathematical analysis of the problem and its extensions is lengthy enough to make a book.

[T]here is enough implied mathematics in the action to make it interesting to professional mathematicians.

It was surprising to learn that the “simple” problem of the Tower of Hanoi and several logical variants contain enough advanced mathematics to make a book like this and could be the subject of a full semester special topics course in advanced mathematics.

Charles Ashbacher, The Mathematical Association of America

[T]his is a book about recreational mathematics, but do not be mistaken, the origin of the questions may be recreational, the problems discussed are undeniable mathematics.

The authors use a very pleasant and amusing style, but they keep the discussion to the point, and leave much more to be explored using many pointers to the extensive reference list. The many illustrations make the technicalities more easy to digest.

Thus if you love puzzles, and more in particular the mathematics behind it, this is a book for you. That holds for mathematicians but for computer scientists as well. Also if you are looking for a lifelasting occupation, then you may find here a list of open problems that will keep you busy for a while.

Adhemar Bultheel, The European Mathematical Society