European Mathematical Society - uroš milutinović
https://euro-math-soc.eu/author/uro%C5%A1-milutinovi%C4%87
enThe Tower of Hanoi - Myths and Maths
https://euro-math-soc.eu/review/tower-hanoi-myths-and-maths
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The <em>Tower of Hanoi</em> (TH) is a classic puzzle that was invented by <em>Éduard Lucas</em> (1842-1891) using the alias <em>N. Claus (de Siam)</em>, an anagram of <em>Lucas d'Amiens</em>. Given three pegs, two of them empty, and a pile of disks of decreasing size is stacked on the third one. The problem is to move the stack to another peg, moving one disk at a time from one peg to another, but never putting a larger disk on top of a smaller one (<em>the divine rule</em>). The <em>Chinese ring</em> puzzle (CR) and the topologically equivalent <em>Cardano rings</em> are related. In the latter two the problem is to get all the rings off the upper bar or to remove the string from the linked pillar structure. (See image below.) These puzzles are older and CR, which Lucas called <em>Baguenaudier</em> puzzle, has probably inspired him to formulate the TH puzzle.</p>
<p>
<img alt="TH puzzle" src="http://www.euro-math-soc.eu/sites/default/files/c031k.jpg" /></p>
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So this 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. There is a lot of graph theory, automata, group actions, complexity analysis and algorithmic aspects around with definitions, theorems and proofs.<br />
Chapter 0 gives an introduction to the problem and the history of the TH puzzle and other related puzzles, but it also introduces definitions and properties of graphs that are used in solving these problems.<br />
Then chapter 1 discusses the CR puzzle as a prototype to set up a model for approaching this kind of problems. The classical TH problem is then much more thoroughly discussed in chapter 2. Algorithms and proofs are given for optimal (w.r.t. the number of steps) solutions. The simplest problem is to move a perfect pile from one peg to another one following the divine rule. All possible intermediate states are called regular. It is more difficult to transform an arbitrary regular state into another one. To solve this, a good understanding and analysis of the associated graph becomes more important.<br />
Lucas considered also another problem where the disks are placed randomly on the three pegs, and the goal is still to transform this irregular state to a perfect or regular state still following the divine rule. That's a short chapter 3.<br />
The next chapter introduces another type of graphs to analyse the TH problem. They are called Sierpinski graphs because of their relation with the Sierpinski triangle.<br />
Generalizing TH from 3 pegs to 4 or more pegs, still looking for an optimal solution (known as Reve's problem), goes way beyond a trivial step. So there is much intuition, conjectures and open problems to be found in chapter 5.<br />
Some further variations are briefly discussed in the next 3 chapters. like more towers (of different color) and more pegs, and/or relaxing the rules of the game (chapter 6) or <em>Towers of London</em>: colored balls on a number of pegs to be brought form one state to another in a minimum number of steps (chapter 7) or with oriented disk moves (chapter 8).<br />
Chapter 9 is just a list of open problems that were encountered in the previous chapters.</p>
<p>
Chapters 0-8 are followed by a list of exercises (hints and solutions are provided in an appendix). In principle the mathematics needed are all introduced in the text, but the introduction to the necessary concepts is kept to a minimum and some basis is tacitly assumed known. For example, the notion of group is used without much ado. Hence a certain mathematical background and some familiarity with graphs is advisable and even required if you want to solve the exercises. Besides the usual subject and name indexes, the glossary of terms with one-liner definitions is quite useful to recall a concept of a previous chapter if you are not so familiar with graphs. 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.</p>
<p>
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.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Image icon" title="image/jpeg" src="/modules/file/icons/image-x-generic.png" /> <a href="https://euro-math-soc.eu/sites/default/files/c031k.jpg" type="image/jpeg; length=20149" title="c031k.jpg">The Tower of Hanoi - Myths and Maths </a></span></div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The mathematics of the classical <em>tower of Hanoi</em> puzzle with 3 pegs and <em>n</em> disks is by now well known and optimal solutions have been described in the form of algorithms that need a minimal number of moves. Besides some historical notes and an introduction to the <em>Chinese ring</em> puzzle, also called <em>Baguenaudier</em> puzzle, the graphs, the number sequences, and in general, the mathematics and algorithms are explained for the <em>tower of Hanoi</em> puzzle. As far as they are currently known, also the methods for its possible generalizations and variants are given. The latter leaves many challenging open problems. Exercises are provide after each chapter.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/andreas-m-hinz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">andreas m. hinz</a></li><li class="vocabulary-links field-item odd"><a href="/author/sandi-klav%C5%BEar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">sandi klavžar</a></li><li class="vocabulary-links field-item even"><a href="/author/uro%C5%A1-milutinovi%C4%87" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">uroš milutinović</a></li><li class="vocabulary-links field-item odd"><a href="/author/ciril-petr" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ciril petr</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0236-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">47.65 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">350</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/978-3-0348-0236-9" title="Link to web page">http://www.springer.com/978-3-0348-0236-9</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/05-combinatorics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05 Combinatorics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/05c90" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05c90</a></li></ul></span>Sat, 23 Feb 2013 08:02:01 +0000Adhemar Bultheel45494 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/tower-hanoi-myths-and-maths#comments