© 2018

The Tower of Hanoi – Myths and Maths


  • Updated Edition of the first comprehensive monograph on the topic

  • Contains new material and a thorough presentation of the historical development

  • Numerous attractive figures and original photos

  • Connections to various mathematical fields and applications to fields like computer science and psychology

  • Exercises with hints and solutions

  • No special knowledge of advanced mathematics assumed from the reader


Table of contents

  1. Front Matter
    Pages i-xviii
  2. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 1-69
  3. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 71-91
  4. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 93-163
  5. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 165-174
  6. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 175-206
  7. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 207-282
  8. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 283-300
  9. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 301-313
  10. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 315-354
  11. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 355-400
  12. Andreas M. Hinz, Sandi Klavžar, Ciril Petr
    Pages 401-404
  13. Back Matter
    Pages 405-458

About this book


The solitaire game “The Tower of Hanoi" was invented in the 19th century by the French number theorist Édouard Lucas. The book presents its mathematical theory and offers a survey of the historical development from predecessors up to recent research. In addition to long-standing myths, it provides a detailed overview of the essential mathematical facts with complete proofs, and also includes unpublished material, e.g., on some captivating integer sequences. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms, together with their correctness proofs, form an essential part of the book. In view of the most important practical applications, namely in physics, network theory and cognitive (neuro)psychology, the book also addresses other structures related to the Tower of Hanoi and its variants.

The updated second edition includes, for the first time in English, the breakthrough reached with the solution of the “The Reve's Puzzle" in 2014. This is a special case of the famed Frame-Stewart conjecture which is still open after more than 75 years. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike.

Excerpts from reviews of the first edition:


“The book is an unusual, but very welcome, form of mathematical writing: recreational mathematics taken seriously and serious mathematics treated historically. 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.”

Chris Sangwin, The Mathematical Intelligencer 37(4) (2015) 87f.


“The 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.”

László Kozma, ACM SIGACT News 45(3) (2014) 34ff.


“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.”

Jean-Paul Allouche, Newsletter of the European Mathematical Society 93 (2014) 56.


Chinese Rings Frame-Stewart conjecture Gray code History of puzzles Sierpiński triangle Tower of London algorithms cognitive tests combinatorics finite automata integer sequences

Authors and affiliations

  1. 1.Faculty of Mathematics, Computer Science and StatisticsLMU MünchenMunichGermany
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematUniversity of MariborMariborSlovenia

About the authors

Andreas M. Hinz is Professor at the Department of  Mathematics, University of Munich (LMU), Germany. He has worked at the University of Geneva (Switzerland), King's College London (England), the Technical University of Munich (Germany), and the Open University in Hagen (Germany). His main fields of research are real analysis, the history of science, mathematical modeling, and discrete mathematics.

Sandi Klavžar is Professor at the Faculty of Mathematics and Physics, University of Ljubljana, Slovenia, and at the Department of Mathematics and Computer Science, University of Maribor, Slovenia. He is an author of three books on graph theory and an editorial board member of numerous journals including Discrete Applied Mathematics, European Journal of Combinatorics, and MATCH Communications in Mathematical and in Computer Chemistry.

Ciril Petr is a researcher at the Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia.

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