European Mathematical Society - 00a08
https://euro-math-soc.eu/msc-full/00a08
enTrigonometric Delights
https://euro-math-soc.eu/review/trigonometric-delights
<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>In the first two sentences of the preface, Maor writes</p>
<blockquote><p>
This book is neither a textbook of trigonometry —of which there are many— nor a comprehensive history of the subject, of which there are almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences.
</p></blockquote>
<p>I could not think of a better characterisation of the book than this. All I can add to this description is to give an idea of which kind of topics were selected and what kind of applications have benefited from these developments.</p>
<p>The successive chapters are organised more or less chronologically, starting with a prologue about the Egyptian Rhind papyrus from around 16-17th century B.C. and ending with Fourier series (18th century). There is of course much attention for the history, but what strikes me in particular, is how much attention is given to the etymology of the mathematical terminology. The origin of the words algorithm and algebra is described in several publications as originating from the Arab author al-Khwarizmi and from al-jabr, which is part of the title of his book, but what is the origin of words such as sine, secant, and many other common mathematical words? Maor carefully pays attention to this. He also shows how trigonometry, which originally was about angles like in pyramid building problems in Egypt, were somewhat made more abstract, in a geometric context of triangles by the Greek, but later, it became more and more part of analysis. The sine and cosine were not only tabulated for computational purposes, but they became functions so that now we see x in sin x as a real or even a complex number, not necessarily corresponding to a physical or geometric angle. The original idea of an angle in degrees or radians in the goniometric unit circle has become somewhat obsolete.</p>
<p>But of course it all starts with angles and chords in planar circles for the Greek, and even earlier in astronomy, which is essentially a three dimensional spherical discipline as practised by Babylonians and almost any civilisation of antiquity. This is the subject of the first two chapters. Then appeared tables of goniometric values of what became our basic goniometric functions. This opens the possibility to introduce algebra (goniometric identities) and gradually also analysis (involving series) into the discipline. This helped considerably to discover (actually re-discover) the heliocentric interpretation of our solar system and to measure our own planet by triangulation and those practical problems in turn stimulated the development of associated trigonometric identities in triangles. But before the heliocentric model, the trajectories of the planets required also more general curves than ther circle like epi- and epo-circles which allow an easy description in terms of trigonometric formulas. Then Maor ventures into a period of proper analysis with the Sine integral and many other relations and series expansions, not in the least for the fascinating number π. These were obtained by master minds such as Gauss and Euler. As complex numbers entered the picture, with Euler's fabulous formula, we are fully involved in complex analysis, conformal maps and ultimately Fourier analysis.</p>
<p>This marvelous survey by Maor of some episodes in the historical evolution of mathematics also allows to sketch some biographies of important mathematicians. There are the "usual suspects" from Greek antiquity (including Zeno whose paradoxes are discussed when infinitesimals from analysis are introduced). Also Regiomontanus (15th C.), François Viète (16th C.), De Moivre (17th C.), Maria Agnesi and her "witch" (18th C.), Jules Lissajous (19th C.), Edmund Landau (20th C.) are discussed in somewhat more detail. aot only the history and mathematicians, also the applications are well documented: astronomy, cartography, spirographs, periodic oscillation, music; and there are detailed mathematical derivations of several trigonometric and other mathematical identities, conformal maps, series converging to π, the solution of the Basel problem by Euler, how Gauss showed that any trigonometric summation formula can be represented geometrically, etc.</p>
<p>All these items are treated requiring only some elementary trigonometric formulas. Some of the standard identities are collected in appendices. In another appendix we find Maor's plea to re-introduce the unit circle and the geometric definitions of the trigonometric functions like cos and sin being projections of the circular point on x- and y-axis, etc. instead of the "New Math" approach. Also Barrow's integration of sec x is moved to an appendix. All the chapters are completed with a section containing notes and references to the sources used. There are many useful mathematical graphs and some grayscale images.</p>
<p>This is an interesting mixture of mathematical history, illustrating the evolution and the usefulness of trigonometry throughout the centuries, and on top of that, it gives some mathematical training by deriving formulas and identities that are easily accessible with only some elementary mathematics knowledge. The book appeared originally in 1998 and is here reprinted in its original form as a volume in the Princeton Science Library. So this is a fortunate occasion to bring this great book back under the attention of a broad audience.</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">Adhemar Bultheel</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>This a reprint in the Princeton Science Library of the original book from 1998. Maor sketches several episodes on the history of mathematics where especially trigonometry was involved from the Rhind Papyrus to Fourier analysis. The history, the mathematicians, the applications, as well as the derivation of mathematical identities are discussed.</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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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/princeton-university-press-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</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">2020</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">9780691202198 (pbk), 9780691202204 (ebk)</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">USD 17.95 (pbk)</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">256</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights" title="Link to web page">https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li></ul></span>Sat, 16 May 2020 14:47:19 +0000Adhemar Bultheel50783 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/trigonometric-delights#commentsThe Tenth Muse
https://euro-math-soc.eu/review/tenth-muse
<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 opening of this novel tells the story of the tenth muse who refuses to be a muse like her nine sisters. She wants to sing her own song, rather than be an inspiration for others, and as a consequence, all her powers are taken away from her and she has to live as a human. She is reincarnating as the women that stood out in the course of history as an artist, philosopher, or, in the case of this novel as a mathematician, who had to stand their ground in a world dominated by men.</p>
<p>
The whole story is told by its main character Katherine, an older successful mathematician, who is reflecting upon her childhood when she grew up in the 1940-50's in Michigan where she showed to be highly gifted with mathematical skills. The main part of the novel is happening while she is entering the university and later as she is working towards a PhD on the (fictional) Mohanty problem that is supposed to be a main opening towards the proof of the Riemann hypothesis. An old (again fictional) Schieling-Meisenbach theorem produced by mathematicians from Göttingen in the 1940's seems to be an essential element to solve the problem but it is also a key element in the novel. Moreover she is going through an identity crisis, and is looking for her, probably German, roots, These two reasons make her decide to spend some time in Bonn (Germany) where she hopes to find what happened to her parents around the time that she was born. Eventually, at the end of the novel, it turns out that Katherine has turned away from (analytic) number theory and has found new challenges in the realm of dynamical systems which has applications in many diverse applied sciences, which is the opposite of what Hardy claimed about number theory.</p>
<p>
The author Catherine Chung has a mathematical degree from the University of Chicago but she received her Master of Fine Arts from Cornell University. For the mathematics in this novel she found some inspiration in popular science books and had some help from friends to check the mathematical content. And there is indeed a lot of mathematics mentioned. As a child, Katherine has to sum the numbers 1 to 9 and immediately comes with the answer 45 using the same technique as it is told that Gauss did by pairing numbers symmetrically in the sequence. She is however not praised for her ingenuity but punished instead because she thinks this is all too obvious and does not want to write down anything. In this case the teacher is a women, but for the rest of the novel, the bad guys are all men. She later goes to college where she is betrayed by her best friend. When they both hand in the same answers to their assignment problems, she is automatically accused of plagiarism while it happened the other way around. More depressing affairs happen to her again and again, even her thesis advisor, with whom she has an affair, disappoints her, and it are always men who do this to her. She is the underdog in all situations: she is not only a women, she is Chinese-Caucasian in the Midwest, is on bad terms with her step-mother, she turns out to be Jewish as well, and she is trying to make a career in science, typically dominated by men. How many stereotypes can be bestowed upon one person. And it is not only happening to her, it happens to most of the women in this novel. When during the war Japanese soldiers threaten to kill the boys of a Chinese family, the daughter is sold to spare the life of the boys. Men think to own a woman and that she can be given away. Even if it is done with the best of intentions, to safe the woman, it still is disrespectful.</p>
<p>
The reason for reviewing this novel here is that the main character is a mathematician, so there is necessarily some mathematics involved. Obviously the Riemann hypothesis appears, but also the zeta function, the Hilbert problems, and the Boltzmann equation show up but without technical details, and some short sections are included about Hypatia, Emmy Noether, Sophie Germain, Sofia Kovalevskaya, Mary Mayer, Ramanujan, Turing, and what happened in Göttingen during the war also plays a role. In a postscript Chung mentions 30 more names of real mathematicians that make a short appearance in the novel: from Gödel and Poincaré to Selberg and Weyl. The link with fiction is made via a fictional Schieling-Meisenbach theorem. Presumably this theorem can be used to solve the equally fictional Mohanty problem (Chandra Mohanty is in real life a professor at Syracuse University defending transnational feminism). What is well described is the urge of a researcher to find a solution for his or her problem in a competing environment where it is important to be the first to publish and one has to be careful not to share too much before, while collaboration is necessary. As a woman or a student this makes you vulnerable because you will always be in the shadow of the co-author. The reader is however not really informed about the details or technicalities of Katherine's research or of most of the mathematics involved, there are only simple descriptions of what topology is about, or some similar descriptions of other topics. Nevertheless, it is instructive for an outsider to learn that when somebody can solve the Mohanty problem for even integers, then there is immediately the challenge of solving the problem for the more difficult case of odd integers. This is how mathematics is a never ending story. The more we know, the more there is left to investigate. Of course the novel is not about the mathematics, but about Katherine who happens to be doing research on a mathematical topic.</p>
<p>
Mix all these issues: mathematics, romantic involvements, treason, the atrocities of the war, gender and race discrimination, and some Buddhist symbolism and Roman mythology and it seems like an overdose that is impossible to concur in one fictional character. And yet, Chung wonderfully succeeds in making it somehow acceptable. It is an engaging story with many twists that keep surprising the reader and that pushes the reader forward, eager to find out what will happen next. There are a lot of mathematical issues that are not revealing much, but still it is remarkable how much of (popular) mathematical issues have been smuggled in by Chung. All of these will be easily assimilated by many readers who would otherwise not be interested in picking up a book about popular mathematics.</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">Adhemar Bultheel</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>
This is a novel about a female mathematician who tries to stand her ground in a world dominated by males. She is like the tenth muse who wants to sing her own songs rather than be the inspiration for others. She also is looking for her identity and in her quest she spends some time in Germany where she learns the truth about her parents and the faith of scientists and mathematicians from Göttingen in the 1940's.</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/catherine-chung" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Catherine Chung</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/harpercollins-ecco" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">HarperCollins /Ecco</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">2019</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">978-0-0625-7406-0 (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">USD 26.99 (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">304</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://www.harpercollins.com/9780062574060/the-tenth-muse/" title="Link to web page">https://www.harpercollins.com/9780062574060/the-tenth-muse/</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 09 Sep 2019 14:01:29 +0000Adhemar Bultheel49706 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/tenth-muse#commentsThe Mathematics of Various Entertaining Subjects volume 3
https://euro-math-soc.eu/review/mathematics-various-entertaining-subjects-volume-3
<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 third of the biennial MOVES (Mathematics of Various Entertaining Subjects) conferences was organized in 2017, as usual in the MoMath museum in New York. This book contains its proceedings. Also <a href="/review/mathematics-various-entertaining-subjects-volume-2" target="_blank">volume 2</a> was reviewed here earlier. The editors, the publisher, and the concept of the previous volumes did not change. The overall theme of the conference was this time <em>The magic of math</em> but that can of course include about anything. There are four parts, each containing three to six papers: (1) Puzzles and brainteasers, (2) Games, (3) Algebra and number theory, (4) Geometry and topology. I will pick some examples to give an idea about the contents.</p>
<p>
The six papers in the part <em>Puzzles and brainteasers</em> usually start by challenging the reader with a number of problems for which the solution is given at the end. An example: a variant of a well know prisoner's hat riddle can be formulated as follows: in a set of prisoners each prisoner gets either a red or a yellow hat. They see the colour of the hats of all the others but they do not know the colour of their own hat. They have to answer simultaneously either the colour of their own hat or pass. If one guesses the wrong colour or if all pass, their sentence is extended. If one guesses correctly and none is wrong, they are freed. What should be their strategy (supposing they want to be free)?. To solve this problem the colours are represented as 0 or 1. Each prisoner can compose the 0-1 string of the ordered set of prisoners, except for his own bit. This can be treated like an error correcting code problem. There exists a set such that the probability that the binary string belongs to that set is much larger than that it belongs to the complement of that set. So the problem is reduced to computing a Hamming distance. The solution is explained in elementary steps, but it eventually ends with explaining Hamming codes, finite fields, parity check matrix, Hamming distance, etc. Another guessing problem involves random walks and hidden variables. So, this shows that what starts as a puzzle or game, will eventually lead to the introduction of some mathematics, which is the set up of almost all the contributions in this book.</p>
<p>
In the <em>Games</em> section we find five papers. Intriguing questions, sometimes with surprising answers, can be asked. Take for example the following ones. What are the chances to have a winning row, column of diagonal in a Bingo game (American style) played with 5 x 5 cards with or without the central square free? How to code and count different Tsuro cards? How difficult is it to loose a checkers game? (Suppose you want your son to win without violating the rules.) Here is another problem involving probability: Each player has to move in turn a random number of steps along a path of squares and the target is to end exactly at the end square. If they don't, they have to turn back on the path until they have made the required number of steps. How many moves are needed on average to finish? How many times is a square visited? Several versions exist of another game called "Japanese ladders" (also known as "Ghost Leg", or with several other names). It is a challenging problem to find a strategy to play these games by adding rungs or legs and win. The mathematical equivalent is to decompose and manipulate permutations as a succession of adjacent transpositions. Answering all these questions in the Games section, involves combinatorics, probability, symmetry, graphs, etc.</p>
<p>
The mathematics required in the part on <em>Algebra and number theory</em> is obvious from its title. The first long paper is by Persi Diaconis and Ron Graham. Diaconis is a well known mathematician and magician and he was an invited speaker at the 2017 MOVES conference. The contribution is about the magic of Charles Sanders Peirce, known to be the father of pragmatism. Peirce's 1908 paper <em>Some amazing mazes</em> is difficult to read, so here Diaconis and Graham analyse the principles that support one of the most complicated card tricks ever, each of these principles can be inspiration for some card trick in its own right. The mathematics involved is interesting as well. The analysis contains for example an implicit proof of Fermat's little theorem. Other papers in this section also involve symmetry, groups, modulo arithmetic, and graphs, to solve games like Khalou, or puzzles like KenKen.</p>
<p>
The three papers in the last part are grouped under the title <em>Geometry and topology</em>. One paper is about flexagons, knots, and twisted bands (of which the Moebius band is the simplest example). The second is an interesting graph problem. Consider a regular triangular grid graph covering the plane with cities located on some of the vertices. What is the shortest set of railroad tracks that allows to reach every city from any other city if the railroad can only move along grid lines? This problem is originating from a board game called TransAmerica, but it is also a practical problem to wire the lights on the grid points of the glass dome of the Dalí museum in St Petersburg, Florida which is constructed as a triangular metal grid frame filled with 1100 triangular windows of approximately equal size (although no two are identical). The last paper in this section is about the incredible amount of ways in which a set of Lego blocks can be connected. Just 8 simple jumper plates (one notch at the top and three slots at the bottom) allow for 393314 different compilations. What is the entropy, i.e., hat are the asymptotics of $\frac{1}{n}\log N(n)$ as $n$ becomes large and $N(n)$ is the number of ways to compile $n$ elements?</p>
<p>
As this incomplete survey illustrates, this is a mixture of fun and serious mathematics where professional mathematicians, computer scientists, and enthusiastic gamers and puzzlers can meet. Recreational mathematics has grown out of its infancy and there are some tough results that can be proved and some serious challenges that can be formulated as open, yet unsolved, problems. Some games and puzzles may have been invented by smart amateurs for recreation, solving the puzzle, winning the game, or computing your chances, has become a topic that often requires some mathematical training. Anyone from amateur to professional will be fascinated by the diversity of challenges and solutions proposed. Not the highest level of mathematical abstraction is needed, so with some elementary knowledge the book can we assimilated, but still, it requires a certain willingness to wade through all the mathematics, which is intended to be an essential part. But mathematics is fun and the book is playful and accessible. Moreover, as Bhargava writes in his foreword: isn't most, if not all, mathematics recreational?</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">Adhemar Bultheel</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>
This contains proceedings of the third biennial MOVES conference (Mathematics of Various Entertaining Subjects) organized in 2017 at the MoMath museum in New York. The papers do not only present the games and puzzles and their fun-aspect, but they connect them to the underlying mathematics that is not always elementary. This is recreational mathematics taken seriously as a mathematical subject.</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/jennifer-beineke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jennifer Beineke</a></li><li class="vocabulary-links field-item odd"><a href="/author/jason-rosenhouse" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jason Rosenhouse</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2019</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">9780691182575 (hbk), 9780691182582 (pbk), 9780691194417 (ebk)</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"> £ 97.00 (hbk), £ 40.00 (pbk)</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">352</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://press.princeton.edu/titles/14228.html" title="Link to web page">https://press.princeton.edu/titles/14228.html</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 09 Sep 2019 13:51:55 +0000Adhemar Bultheel49705 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-various-entertaining-subjects-volume-3#commentsThe Soma Puzzle Book
https://euro-math-soc.eu/review/soma-puzzle-book
<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 classical <a href="https://en.wikipedia.org/wiki/Soma_cube" target="_blank">Soma cube</a> is a 3D puzzle invented by the Danish polymath Piet Hein in 1933. A 3 x 3 x 3 cube is partitioned into seven building blocks. Each of these blocks consists of three or four atoms (that are 1 x 1 cubes) glued together on matching faces and they have at least one inside corner. One block has three atoms (this is called the V block and consists of a corner atom with two atoms glued to two adjacent faces). All the others have four atoms, and are obtained by adding a fourth atom to the V. Three of the these 4-blocks are "flat" (the L the S and the T where the fourth atom is added in the same plane as the V) and three where the fourth atom is added on top of the V outside the V-plane: The P (the fourth atom is on top the corner atom of the V) and the remaining ones (A and B) are on the other blocks of the V (these are left and right chiral). There exist commercial versions of 4 x 4 x 4 or 5 x 5 x 5 cubes for the diehards, but these will not be considered here.</p>
<p>
There are 240 different ways to put the seven pieces together to form the 3 x 3 x 3 cube. The mathematical background has been fully analysed by John Horton Conway in the <em>Mathematical Games</em> column of <em>Scientific American</em> way back in 1958. So this cannot be the subject of this book. What is presented are problems (and solutions) of what other kind of challenges can be posed using these same building blocks. Even with one block there are problems to solve like which block can give a hexagonal shadow or how small can a hole in a plane be that allows to get all (or some of) the pieces through, or what is the largest hole that can be filled with every piece.</p>
<p>
And then the book continues chapter by chapter posing problems with 2, 3,..., 7 pieces. One or two shapes have to be constructed using a selection of building blocks (possibly with duplicates). Also the chapter involving all the seven blocks adds to the classic problem by asking to construct the cube with constraints or to find non cubic volumes. A nice proposal is to construct fractions where a fraction p/q means that there is a bottom layer of q atoms and a second layer of p atoms resting on the blocks of the bottom layer. One can then construct for example fractions 3/9 and 6/9 (whose sum is 1) and there are other such fractions that sum to 1.</p>
<p>
Note that the 3-block with 3 atoms put in one line (that is block I) is excluded and also two 4-blocks (4 in a row which does not fit in the cube and a 2 x 2 square called O) are excluded. If we add the I and O to the set of seven, then new problems can be added to the already extensive list of problems that will now involve 8 or 9 blocks.</p>
<p>
There is no mathematical analysis in this book. A challenge is just graphically presenting the blocks that can be used and the required result. A colourful graphical language is defined that is used in the solution sections to explain how to generate the solution in several steps. This it is a book purely for the fun of solving puzzles. It is of course possible to solve the puzzles with pen and paper if one has a well developed 3D imagination, but it is of course the intention that you have a set (sometimes two sets) of seven blocks physically available, which can be bought in most toy shops. These often already come along with non-cube shapes that have to be built. The current book will add new problems to the existing ones. If you happen to have already such a set, then this book will provide new challenges for you.</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">Adhemar Bultheel</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 book presents new challenges using the seven pieces of the classical Soma cube. The problems and the solutions are presented using colourful graphics. No mathematical knowledge is assumed and no mathematics are explained.</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/david-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li><li class="vocabulary-links field-item odd"><a href="/author/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</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">2019</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">978-981-3275-31-7 (hbk), 978-981-3275-94-2 (pbk), 978-981-3275-33-1 (ebk)</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">£ 40.00 (hbk), £ 25.00 (pbk), £ 19.95 (ebk)</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">180</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://worldscientific.com/worldscibooks/10.1142/11130" title="Link to web page">https://worldscientific.com/worldscibooks/10.1142/11130</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span>Mon, 05 Aug 2019 10:17:32 +0000Adhemar Bultheel49606 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/soma-puzzle-book#commentsThe Paper Puzzle Book
https://euro-math-soc.eu/review/paper-puzzle-book
<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 subtitle of the book : All you need is paper (and scissors and sometimes adhesive tape if you want to be picky), might be tricking you into an imaginary situation of a kindergarten with children producing some artwork for mom, dad, or one of their grand parents. This is a completely different kind of book. You need a well trained set of brains and a strong puzzler's attitude to solve the puzzles that are collected by some of the best.</p>
<p>Ilan Garibi is an Israeli origami specialist, David Goodman is a designer of (mechanical) puzzles, and Yossi Elran is a mathematician, head of the Davidson Institute Science Education Accelerator of the Weizmann Institute in Rehovot, and a big puzzle fan. When they met at a meeting of recreational mathematics and games, the idea for this book was born.</p>
<p>In the best of Martin Gardner's tradition 99 puzzles are collected. Some are classics, some are found in the literature, and others are new. The authors are kind enough to give the origin of the puzzles when appropriate. The number of 99 is just a rough indication because there may be 99 problems formulated, but their solutions, which are given at the end of the chapters, sometimes propose variations or end with an extra challenge left open for the reader.</p>
<p>It may seem not very easy to represent with a static image (or images) in a book, all the necessary operations of folding an cutting that have to be performed in 3D and that sometimes even result in a 3D object. However the different steps are represented using some pictoral vocabulary that is explained in the beginning and that is remarkably clear and easy to read.</p>
<p>The puzzles are grouped according to techniques and topics in ten chapters. Sometimes puzzles are sequential, i.e., you first need to solve puzzle x before you solve puzzle x+1 because solving x is a subproblem of x+1. The puzzles are also rated with one up to four stars. Sometimes the shape of the paper is important for the technique to work: it need to be square or A4, but in other cases it can be just rectangular, or it has to be a long strip. Here is a list of the chapters with some simple illustrative example:<br />
1. Just folding. For example fold a square paper into an equilateral triangle with a follow-up problem to fold the largest possible equilateral triangle that is contained in the square.<br />
2. Origami puzzles. These need so called Kami paper whose sides have different colours, for example black and white. A first exercise is to fold the paper such that the visible areas of black and white are equal. This chapter is rather extensive.<br />
3. 3D folding puzzles. Given a strip of size 1 by 7, fold it into a cube with side 1.<br />
4. Sequence folding. Here one is given for example a square paper with a 2x2 grid defining 4 squares that are marked with the numbers 1 to 4 in lexicographical order. The problem is to fold the paper until it has size 1x1, but such that the squares on the folded stack have the natural order 1,2,3,4. Many variations are possible, starting from different configurations, or allowing a few cuts, etc.<br />
5. Strips of paper. Here of course the Möbius band plays a prominent role, but there are other puzzles to formulate with strips.<br />
6. Flexagons. This is an invention of Artur Stone of 1939 and popularized by Martin Gardner and later picked up by several others. Paper is folded into a polygonal form in such a way that that it has a front and a back side, but it allows for an simple flipping operation such that it is so to speak turned inside-out, showing different faces. One could define it as a flat folded configuration that has more than two faces. As a simple example one could start from a particular configuration of 6 connected squares (neighbouring squares have exactly one edge in common). Both sides have two squares marked 1, two marked 2 and two marked 3. Counting both sides, there are thus four 1's, four 2's and four 3's. This has to be folded into a 2x2 square and the 'first' and 'last' square are taped together so that one gets a sort of Möbius ring object that will allow only a limited number of hinged flips. The 2x2 square has to show the four 1's on the front and the four 2's on the back. By 'flipping' it, one gets all 3's on one side and all 2's on the other. There are three faces that can be shown in turn by flipping.<br />
7. Fold and cut. For example, you have to fold a piece of paper in a certain way and cut it with one straight cut to obtain a prescribed shape like a cross or a star.<br />
8. Just cutting. A classic is to cut a hole in an A4 size paper, such that a person can step through the hole without tearing the paper.<br />
9. Overlapping paper puzzles. It is clear that, given three paper squares, one may arrange them in a partially overlapping way such that all three are only partially visible. This is impossible with four squares. Problems based on this principle can be formulated putting restrictions of the number or size of papers you start with, or restrictions on the shape of the outer boundary of the stacked papers.<br />
10. More fun with paper. This is the miscellaneous section with many diverse fun constructs like putting together a rotator or an helicopter, performing magic tricks, solve (seemingly) impossible bets, etc.</p>
<p>The examples I gave above are just to illustrate the idea of what kind of puzzles are possible. They are usually the first kick start puzzles for the chapter rated with one or two stars. Sometimes these innocent looking problems can be be surprisingly difficult to solve even if they get the lowest difficulty rating. Although the solution methods for the puzzles are reminiscent to geometry, no mathematics is required. It reminds me of the ancient Greek idea of constructions using only compass and straightedge, but this is definitely different and even more basic: there is no compass, and there is no ruler. It is for example difficult to divide an edge of a square in three (or in n if n is odd) equal parts. That is only possible using an iterative pinching procedure. Such basic techniques are explained in an appendix. There is also a (limited) list of books, papers, and websites for further reading.</p>
<p>This is a marvellous book. The diversity of possible puzzles that can be given with these very limited resources, which are basically some paper and scissors, is overwhelming, and the challenges are sometimes very tough. Even the two-star problems may be hard for an untrained puzzler. This is medicine against boredom on long rainy days, but be careful not to get addicted or it may suck up your less empty and sunny days as well.</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">Adhemar Bultheel</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>This is a marvellous set of about a hundred puzzles that have to be solved by only folding and/or cutting paper. They were collected by three experts: an origami specialist, a puzzle designer, and a mathematician. Many of these innocent looking problems are really hard to solve, and others seem to be impossible at first sight. It requires geometrical thinking, but no mathematical knowledge is needed. As with many of these mathematical puzzles you need to be able to think outside the box, and sometimes to visualize things in 3D.</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/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</a></li><li class="vocabulary-links field-item odd"><a href="/author/david-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li><li class="vocabulary-links field-item even"><a href="/author/yossi-elran" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yossi Elran</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</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">2018</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">978-981-3202-40-5 (hbk), 978-981-3202-41-2 (pbk), 978-981-3202-43-6 (ebk)</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">GBP42.00 (hbk), GBP25.00 (pbk), GBP20.00 (ebk)</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">264</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://www.worldscientific.com/worldscibooks/10.1142/10324" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10324</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Thu, 10 May 2018 06:28:44 +0000Adhemar Bultheel48455 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/paper-puzzle-book#commentsFoolproof, and Other Mathematical Meditations
https://euro-math-soc.eu/review/foolproof-and-other-mathematical-meditations
<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>
Brian Hayes started his career as a member of the editorial staff of <em>Scientific American</em> in 1973. From 1983 on, his <em>Computer Recreations</em> continued the columns previously written by Martin Gardner and Douglas Hofstadter in <em>Scientific American</em>. After he left <em>Scientific American</em> he was mainly active as a contributor to (and for two years also as the editor of) <em>American Scientist</em>, a bimonthly magazine devoted to science and technology. Two books with selections of his articles appeared before. The present book is a collection of updated and extended versions of 13 of his contributions that appeared previously in <em>American Scientist</em>. The texts as they appeared originally were exposed to a broad readership and so many the reactions and additions could be implemented in the current version.</p>
<p>
The topics covered by these 13 essays, as the author calls them, are diverse. Several of the topics are familiar subjects in popular science writing, but what I appreciate most is how Hayes transfers his interest in the subject to the reader. He is not just transferring knowledge of other authors, collecting the ideas from the literature, but he takes the reader along, showing how he explored the topic himself in his quest to understand the underlying truth or the mathematical law.</p>
<p>
Anyone with an interest in puzzles and/or science and mathematics will love this book. No specific mathematical knowledge is required. To give a flavour of the contents, here is a quick survey.</p>
<p>
The first text investigates the well known story about Gauss, who as a schoolboy had to add the numbers from 1 to 100, and how he surprised the school teacher putting the right answer on his desk with a "ligget se". He had inventing the sum formula for an arithmetic sequence by adding symmetric pairs arriving at 50 pairs whose sum is 101. Just telling this story (adding a personal pinch of drama), is where most authors stop, but it is where Hayes starts. Where does this story come from? How much is authentic? Did Gauss indeed count symmetric pairs? How much time does it take to add the 100 numbers sequentially? And many more similar questions. Hayes concludes after dragging the historical literature that the origin seems to be the biography of Gauss by Sartorius written in 1856 as an obituary for Gauss who died the year before. However the story has been moulded and modified a lot since then. Hayes has compiled and investigated the most complete collection of the different versions of how the story has been told since then.<br />
Another historical detective work is the chapter where Hayes investigates what went wrong when in 1873 William Shanks published 707 digits of pi that he had calculated by hand. However, only the first 527 places were correct as D.L. Ferguson detected in 1944. Like a forensic investigator, Hayes analyses the computations of this cold case to find out which exactly were the errors that Shanks made and succeeds in identifying several omissions in transcribing the numbers.</p>
<p>
Statistics is used in several of the chapters. We all know that to compute an average, we can increase the quality of the estimate by increasing the sample size, unless this process does not converge. A counter example investigated here is the factorial-like function <em>n</em>? To find the function value, one has to keep multiplying randomly selected integers between 1 and <em>n</em> (possibly selecting the same number repeatedly) until one hits the number 1. The function value is then the product obtained so far. Playing with this function, Hayes shows that the average outcome increases with the sample size.<br />
More statistics are used in the chapter on Zeno's game. You and another players start with the same amount of coins. In step <em>n</em> you bet 1/<em>n</em> for the outcome of a fair coin toss. If you win you get 1/<em>n</em> of the other one's budget, it not, you loose 1/<em>n</em> of yours. Suppose you loose the first tosses, can you ever recover from your loss and win in the end? This is of course related to the divergence of the harmonic series, but it can also be connected with Cantor sets and with random walks on binary trees.</p>
<p>
Random walks, space filling curves, and self-avoiding walks have practical applications but they are also fun to play with. For example, when walking on a rectangular grid, how many <em>n</em>-step self-avoiding walks do there exist. No exact formula is known, but an heuristic one has been proposed. The asymptotic behaviour as <em>n</em> approaches infinity seems to give rise to a so called connectivity constant in this formula. Experimental values were obtained, but so far no exact solution has been found. It is a tantalizing challenge to find out whether it is a (simple) expression in terms of known transcendental numbers.<br />
In another chapter space filling curves are defined recursively leading to fractals, Cantor's calculus of the infinite, and to approximate solutions of the travelling salesman problem.<br />
Another counting problem is to figure out the number of different (uniquely defined) solutions for a sudoku of order <em>n</em>. The history of sudokus is briefly discussed and some heuristics are given for solving them. The number of givens does not seem to be a good criterion to classify a sudoku as easy or hard. More mathematical approaches to the problem are found in the references. It is strange that the book by J. Rosenhouse and L. Taalman <em>Taking sudoku seriously</em> Oxford U. Press, 20012, is missing from the list. Anyway the sudoku problem is hard and assumed to be NP complete. The proof that 17 is the minimum number of givens for the usual order 3 sudoku was only proved in 2012 by exhaustively checking all the approximately 5 billion solutions.<br />
Markov chains is another statistical subject. Finding the probability of a letter following a group of letters or words following a group of words, can be characteristic for the text produced by a particular author. With these probabilities, one may write a computer program that will generate (gibberish) text in the style of that author.</p>
<p>
The distribution of discrete random variables are caught in a spectrum, like the eigenvalues of a random matrix, or the zeros of the Riemann zeta function or seismic activity, etc. Hilbert and Polya conjectured that there is some universal operator whose spectrum corresponds to the distribution of the Riemann zeta zeros. This can be interpreted as corresponding to the jumps in the energy levels of the nucleus of some imaginary chemical element that could be named Riemannium.<br />
The distinction between pure random numbers and the more advantageous quasi- and pseudo-random numbers in computer applications is clearly explained in a chapter where it is also explained how Monte Carlo-type methods can solve high dimensional integration problems.</p>
<p>
It is a known paradox that volume of the largest ball that can be inscribed in the <em>n</em> dimensional unit cube when <em>n</em> is large is surprisingly small. Hayes gives arguments that convince the reader that this is not so surprising, but that it should actually be an obvious fact.<br />
Sometimes Hayes includes short computer code snippets that he used in his experiments. One chapter is devoted to the representation of floating point numbers on computers. Assigning a fixed finite number of bits to represent a floating point number results in finite precision and rounding errors that accumulate during the computations. This may have catastrophic consequences if not kept under control. Besides the standard IEEE representation, alternatives exist that are more flexible in distributing the available bits for representing a number between the significant and the exponent.</p>
<p>
The final chapter is called <em>Foolproof</em> which is also the title of the book. It sketches how the concept of a mathematical proof has evolved since Socrates. Nowadays computer proofs are more common, but the first computer assisted proof (of the four colour problem in 1976) had a hard time to get accepted. But also proofs provided by humans can be very long or perhaps very cryptic so that it takes years to check them. The incentive for this chapter is the proof by Wanzel in 1837 that the trisection of an angle using only ruler and compass is impossible. This was known since ancient Greece, but no proof was given until then, and yet it remained in obscurity for quite a while.</p>
<p>
I have tried to give an idea of what the different subjects are without including too many spoilers. There is plenty left to discover and savour for the connoisseur. Hayes has a pleasant style and he is taking you along his personal exploration of the subject. There are ample references provided for those who are hungry for more details. Hayes also refers at several instances to his web site at <a href="http://bit-player.org/" target="_blank">bit-player.org</a> for additional material. These web pages claim to be <em>An amateur's outlook on computation and mathematics</em>. You can find many animations and texts that will also be of interest for the readers of this book.</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">Adhemar Bultheel</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>
Hayes has contributed many articles to the bimonthly magazine <em>American Scientist</em> discussing some mathematical topics for a broad audience. This book is a collection of 13 of these texts that have been updated and polished. Recommended for lovers of popular science and recreational mathematics. Hayes' own explorations are gentle invitations to do similar computer experiments. In this way, skin diving just below the surface of an intriguing phenomenon, you will have the personal satisfaction of discovering some of the hidden mathematical rules, the "why" of what is observed.</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/brian-hayes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian Hayes</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/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</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">2017</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">9780262036863 (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">USD 34.95 (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">248</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://mitpress.mit.edu/books/foolproof-and-other-mathematical-meditations" title="Link to web page">https://mitpress.mit.edu/books/foolproof-and-other-mathematical-meditations</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Fri, 27 Oct 2017 05:41:06 +0000Adhemar Bultheel47967 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/foolproof-and-other-mathematical-meditations#commentsThe Mathematics of Various Entertaining Subjects volume 2
https://euro-math-soc.eu/review/mathematics-various-entertaining-subjects-volume-2
<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>
Recreational mathematics, puzzles and games have by now a long tradition and there are many books, papers, and columns devoted to these topics. Many love the challenges of the puzzles and games and solving the problem is rewarded with a dopamine rush when a solution is finally reached. They love it as a pastime or a hobby. However, solving the problem in a systematic way may require some more serious mathematical research. With the increasing popularity of recreational mathematics new mathematical challenges emerged which transcended the recreational aspect. In fact, historically speaking, some well established branches of mathematics like for example probability theory and graph theory, grew out of trying to solve a recreational problem. Another example mentioned by the editors in their foreword is the development of a theory for combinatorial games which have led Elwyn Berlekamp, John Conway and Richard Guy to compiling their multivolume work <em>Winning Ways for Your Mathematical Plays</em>.</p>
<p>
In 2013 the first MOVES (<em>Mathematics Of Various Entertaining Subjects</em>) conference was organized, which was the start of a series of biennial meetings with an increasing number of participants. This is volume two, a sequel to the book with the same title. The first book can be considered as a kind of proceedings for the 2013 MOVES conference and the present one collects nineteen research papers most of which were presented at the 2015 conference. Berlekamp, Conway and Guy were all three participating as invited speakers then. The difference with the usual books collecting papers about this subject is that here the authors have included the "serious mathematics" behind the recreational aspect. The contributions are organized in five different parts: (1) Puzzles and brain teasers, (2) Geometry and topology, (3) Graph theory, (4) Games of chance, and (5) Computational complexity. Those familiar with recreational mathematics will recognize that most of the problems they know will fall under one (or more) of these headings. One may miss logic as a popular subject, but that falls under the first category. Among the contributors are some well known names:. R. Guy, J. Rosenhouse, P. Stockmeyer, E. and M. Demaine, J. Conway, N. Elkies, and many others.</p>
<p>
In order to give an idea of the type of contributions, let me discuss some examples. I will pick one from each of the five parts. In the part about Puzzles and brain teasers, Tanya Khovanova wrote about <em>Dragons and kasha</em>. The setting is as follows: On each face of a cube sits a four armed dragon with a bowl of kasha. Each minute they grab one fourth of the kasha from the bowls of their four neighbours. Given an initial distribution of kasha in the six bowls, what is the asymptotic distribution? The problem is not difficult to solve and the result is that each dragon will end up with the same amount of kasha, whatever the initial distribution. The nice thing about the paper is that it relates the solution to Markov chains, random walks, eigenvalue problems, but the real purpose is by defining a "stealing operator" and replacing kasha by complex numbers, this can be linked to intertwining operators, group actions, and group representations. The added value of this paper is that such abstract concepts are introduced in an easily accessible and playful way. This is an example of a paper that is playful and does not require much mathematical skills from the reader.</p>
<p>
Jill Bigley Dunham and Gwyneth R. Whieldon have a paper in the Geometry and topology part on counting the solutions to a paper cutting and folding problem proposed by Martin Gardner. Given is a square piece of paper subdivided by a regular 3 x 3 gird. It is black on one side and white on the other. Cutting and folding is allowed only along grid lines, and the cutting should not result in disjoint pieces. The challenge is by cutting and folding to wrap this around a 1 x 1 cube showing black faces on all sides. The problem is not so difficult to solve, but many much more complicated variants can be imagined. The way in which the nine grid squares are connected can be represented by a graph: each 1 x 1 square is a vertex and vertices are connected by an edge when the squares are connected. A cut between two neighbouring grid squares corresponds to removing an edge in the graph. Cutting and folding structures and wrapping sequences can be enumerated and programmed. Because the number of possibilities, given the constraints, is not too large, all possibilities can be checked. By this enumeration, a theorem can be proved: A wrapping solution exists for a cutting pattern if and only if there exists a non-self-intersecting sequence of four moves to adjacent squares starting at the centre square. Some patterns allowed several wrappings, some of them are mono-coloured other mix black and white. Thus a complete enumeration of all possible solutions and a list of cutting patterns that do not allow any solution can be listed by a computer program.</p>
<p>
More graph theory in the paper by Dominic Lanphier from the part on Graphs. Suppose you are attending a sequential duel, which means that duelist 1 aims at duelist 2 and fires. If he misses (with probability 1 − <em>p</em>, then number 2 fires at number 1 and hits with probability <em>q</em>. If he misses, the first duelist fires again etc. The problem is to find out which one has the most chance to survive. The problem can be made more complicate if more than two shooters fire at each other in some order, which could be cyclic or not. What if after one has been shot, the survivors have to continue until there is only one survivor. If you are one of the shooters in a large pool, you better learn the necessary statistics, generating functions and asymptotics discussed in this paper to maximize your chance to survive. It requires some more mathematics and computations than the two previous examples, but it is still very accessible for anyone who would need it. More advanced mathematics is needed for the paper by Noam Elkies from the same part in which the number of crossings in a complete graph is computed. If in a planar graph every vertex is connected to every other vertex, the edges will necessarily cross a minimal number of times. Counting these crossings in not so difficult when it concerns plane graphs, but it is less trivial if these graphs live on another topology like a sphere, a toroid, a projective plane, a Moebius strip,...</p>
<p>
When in part four chance and probability become an essential element in the game, then dice naturally come to the foreground. They literally do in the first contribution of this part authored by Robert Bosch, Robert Fathauer and Henry Segerman. If it is true that for fair dice every face has an equal chance to land on top, then we could number faces in any configuration. In practice however they are always numerically balanced, that is they always have opposite faces numbered (1,6), (2,5) and (3,4). In the paper this phenomenon is investigated for the 20-sided (icosahedral) dice where opposite sides sum to 21. Each of the 12 vertices is common to five triangular faces. Optimal numerical balancing could consider not only the opposite sides, but also the vertex sum (the sum of the 5 adjacent faces), or the face sum (the sum of the 3 adjacent faces) with or without including the number on the face itself, or the sum of the equatorial bands (the sum of the 10 faces on the equator if two opposite vertices are selected as north and south pole). Solving such problems requires integer constrained programming to find a solution and is related to magic squares. Generalisations from d20 to d30 and d120 dice are also considered.</p>
<p>
In the computational complexity part, we find a relatively long paper by Aviv Adler, Erik Demaine and others who give a proof of the fact that Clickomania is a hard problem even if there are only two colours and two columns. The game is a popular computer game that starts with a rectangular grid tiled with coloured squares. With one click you can remove a connected structure of tiles with the same colour. The empty space is filled with tiles higher up in the same column. Isolated singletons can not be removed and empty columns collapse so that their left and right neighbours join. Either the target is to remove all tiles or to get the highest possible score. Each click adds to your score which is about the square of the number of tiles you remove in that click. The theorem says that if the target is to remove all tiles, then this is an NP-complete problem, unless we have the trivial situation of only one column and one colour. The score variant is NP-complete if there is more than one column and more than one colour, and trivial (and hence in complexity class P) if there is only one colour. Proving this theorem requires first of all a speed course in complexity theory to define the terminology and the process of reduction has to be explained, that is how a problem, known to be NP-complete, can be transformed (in polynomial time) into another one, in order to conclude that the latter will be NP-complete too. Given all this basic material, then the proof itself is far from trivial. And if the problem is indeed NP-complete, then one still wants to find a good approximation. Such approximations and variants are also discussed and, for the ambitious reader, there are still many open problems left to be solved.</p>
<p>
It should be clear from this selection that the papers cover not only a very diverse set of problems, but also that they are not always at the same level of complication. Nevertheless all authors have attempted to reach a general public with a certain mathematical training. We could conclude that this is a book on recreational mathematics for the mathematician. This does indeed reflect the audience that attended the MOVES conference in 2015. I suppose these participants will be interested in a copy of this book but even more so those who could not be there but would have loved to. They can still get an idea of what kind of problems were discussed.</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">Adhemar Bultheel</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>
This is number two in what is probably going to be a series of a sort of proceedings of the biennial MOVES conferences (Mathematics of Various Entertaining Subjects) organized at the MoMath museum in New York. The first book was a selection of papers of the 2013 conference, the current one of the 2015 conference. The papers do not only present the games and puzzles and their fun-aspect, but they connect them to "real mathematics". One could characterize them as books on recreational mathematics for mathematicians. </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/jennifer-beineke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jennifer Beineke</a></li><li class="vocabulary-links field-item odd"><a href="/author/jason-rosenhouse" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jason Rosenhouse</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2017</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">9780691171920 (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"> £ 70.95 (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">408</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/control-theory-and-optimization" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Control Theory and Optimization</a></li><li class="vocabulary-links field-item odd"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item even"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li></ul></span><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="https://press.princeton.edu/titles/11171.html" title="Link to web page">https://press.princeton.edu/titles/11171.html</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/05a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05A99</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05c99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05C99</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03d15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/91-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span>Fri, 13 Oct 2017 11:06:35 +0000Adhemar Bultheel47932 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-various-entertaining-subjects-volume-2#commentsStrategy Games to Enhance Problem-Solving Ability in Mathematics
https://euro-math-soc.eu/review/strategy-games-enhance-problem-solving-ability-mathematics
<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>
In the introduction the authors explain the goal of their book. When two opponents are playing a board game they need some strategy to win or at least maximize the chances not to loose the game. Mastering such strategies is a skill that can also be used in solving a (mathematical) problem. They even give a table drawing parallels between playing (and winning) the game versus the steps in solving a problem. I believe this is only superficial and much better and deeper analogies do exist. Winning a game is just another problem, where you have to first understand the goal, learn the rules, and develop some strategy. By repeated playing, one learns which strategies work and most of all which strategies work consistently and are not just a lucky hap. The meta question is then whether this strategy can be generalized, to other problems of larger dimension or when the rules are inverted or only slightly changed etc. This is a general approach to solve any problem, whether the problem is mathematical or not.</p>
<p>
Once this has been made clear, the following chapters are basically an enumeration of games, stating the rules and the goal of the game. There is also what the authors call a "sample simulation" in which some moves or situations are simulated, pointing to some problems or possibilities, making suggestions, and sometimes asking questions to think about.</p>
<p>
The different chapters group games that are similar or just form variations of the same game or at least they have similar goals and thus may require similar strategies. We have a chapter on tic-tac-toe-like games, one chapter is called "blocking games" which can come in many different forms (nim is and example), another chapter deals with games where the strategy has to be continuously updated while playing and finally a "miscellaneous" chapter where, among others, several classic western games are found like checkers, dominoes, battleship,...</p>
<p>
So far, only the problems were formulated and the reader, or rather the players, are supposed to go through the steps that I sketched in the first paragraph: finding and analysing a winning strategy. The last chapter provides answers and hints to the problems in the previous chapters. What are possible strategies? For example what is the best first move to open the game? Sometimes a mathematical proof may exist for the claim that the one who starts (and makes no mistakes) will always win. However there are no mathematics involved here, although there could have been. And these mathematics need not always be connected with game theory.</p>
<p>
In an appendix, illustrations of the (empty) game boards are given, although their geometry should be clear already from the previous chapters. Even if these pages are cut from the book, they may be too small to play on. I do not see the advantage of adding these pages to the printed book. Perhaps an electronic version could be printed with some magnification factor.</p>
<p>
In conclusion, this is a nice collection of board games, and when pupils will play such games, they will develop some winning strategies for these games, and these skills will probably help in cultivating certain attitudes and perhaps working schemes to tackle mathematical problems. However it was certainly not the intention of the authors to involve mathematics in this book. I think however that it would not be very difficult to hook up several mathematical problems to these games, somewhat like what Matthew Lane did for video games in his book <a href="/review/power-unlocking-hidden-mathematics-video-games" target="_blank">Power Up</a>. But that would be a completely different book because here the focus is just the games and learning how to win them mostly by playing them, thereby avoiding all the mathematics and abstract game theory.</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">Adhemar Bultheel</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 author's leitmotif in this book is that when two players are trying to win a board game, they will develop a winning strategy which is very similar to the strategies and skills needed to solve other, more mathematical, problems. So they describe many of these board games with their rules and urge the reader to play these games several times until they see some strategy emerge. In a final chapter they give some hints about what such strategies may look like for each of these games. There are however no mathematics involved as such.</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/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/stephen-krulik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Krulik</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/world-scientfic" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">World Scientfic</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">2017</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">978-981-3146-341 (pbk)</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">GBP 20.00</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">136</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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.worldscientific.com/worldscibooks/10.1142/10187" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/10187</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/91a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91A05</a></li></ul></span>Fri, 23 Jun 2017 11:37:34 +0000Adhemar Bultheel47737 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/strategy-games-enhance-problem-solving-ability-mathematics#commentsThe Canterbury Puzzles
https://euro-math-soc.eu/review/canterbury-puzzles
<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>
Henry Dudeney (1857-1930) was an English mathematician who is best remembered for his logic puzzles and games. He published several books that collected sets of his puzzles that appeared in magazines before and to which he added some new material. <em>The Canterbury Puzzles</em> (1907) was his first collection. The title refers to the fact that the first set of the puzzles are presented as if formulated by characters from Chaucer's <em>Canterbury Tales</em>. Ten years later he had another collection published <em>Amusements in Mathematics</em> (1917). In the 1920's two more collections were published and more came out posthumously.</p>
<p>
The <em>Canterbury Puzzles</em> and the <em>Amusements in Mathematics</em> are two classics and have been available as Dover Publications for some time now. More recently they were also available on ebook repositories such as <em>Project Gutenberg</em> and others. The present book is a pocket edition by Penguin that reproduces a 1919 edition of the <em>Canterbury Puzzles</em> published by Thomas Nelson and Sons. So this edition is from after <em>Amusements in Mathematics</em> was published. Although the two books contain a largely disjunct set, some of the puzzles in the <em>Canterbury Puzzles</em> are related to or are variations of puzzles that also appeared in the <em>Amusements in Mathematics</em>, so for some of the discussions, Dudeney refers in this edition to that book for some extra information or related puzzles.</p>
<p>
The book consists of several chapters in which the puzzles are formulated and at the end of the book the solutions are given. The chapters merely differ by the context in which the puzzles are placed, not by the kind of puzzles they contain. The first chapter has the same title as the title of the book and one should be prepared to read some Chaucer's Middle English. Some of the subsequent chapters are also placed in the same realm of medieval castles and monasteries. Later it moves to a more "modern" decorum (recall though that this is written some hundred years ago at the dawn of the 20th century, which obviously is somewhat reflected in the wording and the style, and certainly in the many illustrations). There are puzzles of all sorts to be found in each chapter, geometric as well as combinatorial or even crime mysteries that have to be solved like in a whodunit mystery. Almost all of the problems are illustrated, not only to formulate the problem or the solution when it is geometric, but also with the scenes in which monks, lords, pilgrims, or other persons are figuring. Also the degree of difficulty is rather diverse. Mathematical education is not really needed, simple counting suffices. The problems primarily rely on creative and careful logic thinking. The keywords in the index that was added to this 1919 edition should make it possible to look up a certain (type of) puzzle among the 114 items that the book contains.</p>
<p>
One of the puzzles is the famous Haberdasher puzzle that was formulated by Dudeney in 1902. The problem is to cut up an equilateral triangle into four pieces that have to be rearranged to form a square. The solution given by Dudeney is in the form of a hinged dissection puzzle. Hinged dissections became later a particular type of puzzles. Gregg Frederickson published a couple of books on this kind of problems. Also Martin Gardner has discussed it and he wrote a tribute to Dudeney in his <em>Scientific American</em> columns. Of course Gardner also published several collections of his puzzles. The Haberdasher problem also features in <em>The Penguin book of curious and interesting puzzles</em> (1992) by David Wells which is another collection of classical puzzles. <em>The Moscow puzzles</em> (1956) by Boris Kordemsky is yet another classic that was very popular in the Soviet Union. It will also be republished by Penguin. And nowadays there are of course many more collections available on the book market of popular science and mathematics.</p>
<p>
It is fortunate that Penguin republishes these classics and makes them available again. Dudeney was one of the first of what has become a flourishing branch of recreational mathematics and logical games and puzzles, a market that is fuelled by traditional puzzle clubs and currently the more trendy math jams. </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">Adhemar Bultheel</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>
This is a classic puzzle book in which part of the problems are placed in a medieval context and where some characters of Chaucer's book do appear. The original is from 1908, but the text reproduced here is from a revision originally published in 1919. It is one of the earliest puzzle books of its kind. <br />
</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/henry-dudeney" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Henry Dudeney</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/penguin-michael-joseph" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Penguin / Michael Joseph</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">2017</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">9780718187088 (pbk)</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">£9.99 (pbk)</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">256</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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="https://www.penguin.co.uk/books/304828/the-canterbury-puzzles/" title="Link to web page">https://www.penguin.co.uk/books/304828/the-canterbury-puzzles/</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span>Fri, 23 Jun 2017 11:26:34 +0000Adhemar Bultheel47736 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/canterbury-puzzles#commentsNice Numbers
https://euro-math-soc.eu/review/nice-numbers
<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>
For readers who are used to read books that promote and popularize mathematics, John Barnes may be a familiar name since he published before <em>Gems of Geometry</em>(Springer, 2009, 2012). What he did for geometry there, he is now doing for numbers, another topic commonly used to reach a broad audience. And audience is to be taken in a literal sense because both books are the result of a series of lectures given by the author.</p>
<p>
There are 10 chapters in this book, corresponding to 10 lectures, each one discussing a general aspect related to numbers. Of course many authors have been excavating this topic with the same ambition before and much of what Barnes tells us has been told by others on several occasions. Yet there is always a fresh angle to look at them and there is always some sparkling gem, something unknown before, or an unexpected link or connection to be discovered.</p>
<p>
A quick survey of the chapters.<br />
<strong>1. Measures.</strong> Defining prime numbers and factors leads to an extensive discussion of many non-decimal systems of units for currency, length, volume, time, weight,... In fact 10 has too few factors and therefore other numbers can be more suitable as a base.<br />
<strong>2. Amicable numbers.</strong> Perfect numbers equal the sum of their factors and amicable numbers is a couple where each one equals the sum of the factors of the other. This can be generalized to n-tuples or sociable cycles. Mersenne primes and Fermat numbers are introduced here. <br />
<strong>3. Probability.</strong> Of course this involves coin flipping and dice throwing and normal and other distributions. But as an aside, it is surprising to learn that there are 16 different possibilities to produce a classical die. Furthermore we recognize the Buffon needle problem and the Monty Hall problem. And there is the practical problem of false positives in tests and more playfully several game strategies and gambling theoretical problems.<br />
<strong>4. Fractions.</strong> This is explaining the Egyptian number system, continued fractions, repeating decimal expansions of rational numbers, and a long division-like algorithm to compute a square root (a forgotten skill now that pocket calculators are generally available). <br />
<strong>5. Time.</strong> This discusses the Roman and the Gregorian calendar, but also the astronomical origins of year, season, month, week, and night and day.<br />
<strong>6. Notations.</strong> Thinking of the notation of numbers, this must obviously include our familiar decimal positional system, but also predecessors like the Roman and the Babylonian system. This is also a good place to look at modular arithmetic (also used in other chapters). The repeating expansion of rational numbers is reconsidered in a different base system and connected with Fermat's little theorem.<br />
<strong>7. Bells.</strong> Bell ringing (or tintinnalogia) is a combinatorial problem in which one has to ring a set of bells with certain patterns of repetition. This will involve decomposition of permutations and even some group theory, although the latter is not elaborated here.<br />
<strong>8. Primes.</strong> This involves classical ideas like the Euclidean algorithm and the sieve of Eratosthenes, but also Gaussian (complex) primes, and prime polynomials with coefficients in modular arithmetic.<br />
<strong>9. Music.</strong> This is a relatively long chapter about the different music scales that can be used.<br />
<strong>10. Finale.</strong> This is a roundup of three remaining topics. The most important is an explanation of the working of the public key RSA encryption algorithm. Furthermore possible animal gaits, and finally the towers of Hanoi and the topologically equivalent Chinese ring puzzle.</p>
<p>
In his lectures, Barnes did involve his public actively, which is reflected by the exercises that are given at the end of the chapters. Easy exercises that do not need extra knowledge than what was explained in that chapter. For those who are willing to read more, some references are provided with some advise about their content and their difficulty.</p>
<p>
So, if you are not familiar with popularizing books using `numbers' as a master key to introduce mathematics, this is an excellent start. It is light, entertaining, richly illustrated, and still Barnes has disregarded the advise of many publishes to avoid all formulas since each formula allegedly would cut the sales in half. I tend to disagree with those publishers and I am happy that Barnes did too. I am sure that those who are willing to read this kind of books are not scarred away by a formula. The problem with formulas is that typos easily slip in, like a blatant one on page 70 claiming that $\sqrt{2}$ is a root of the equation $x^2−1=0$. Even if formulas are allowed, the content should still be digestible for anyone, and it should be easy for some of the readers to skip the more technical parts, while these are still available for those who are hungry for more. In this case there are 100 more pages with nine appendices of such marvelous items. Some are indeed more mathematical yet still entertaining (Ackermann function, Pascal's triangle and 3D generalization of triangular numbers, stochastics in game and queuing theory, the Chinese remainder theorem, group theory and an extensive one on the solution of the Rubik cube, etc.). This book is a most enjoyable read. Those who read <em>Gems of Geometry</em> need not be convinced of Barnes' entertaining style and will love to read this book too. Others who read this book without knowing his geometry book will be teased to also check that one out.</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">Adhemar Bultheel</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>
This book, like Barnes' previous <em>Gems of Geometry</em> is based on lectures given for a general audience. It discusses in 10 chapters and in nine a bit more advanced appendices some topics from number theory accessible to a general readership. There are mathematical formulas and even an occasional proof, but everything is brought in an entertaining style and there are some pleasantly surprising side stories like patterns for bell ringing and non-decimal subdivided units for all sorts of measurements, or 16 different ways to produce a genuine die. </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/john-barnes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Barnes</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-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</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">2016</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">978-3319468303 (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">USD 39.99 (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">329</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><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/gp/book/9783319468303" title="Link to web page">http://www.springer.com/gp/book/9783319468303</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11a51" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A51</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11bxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Bxx</a></li></ul></span>Sun, 19 Feb 2017 09:35:32 +0000Adhemar Bultheel47466 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/nice-numbers#comments