# David Singmaster's books

We give information below about six books by David Singmaster. There are more books to which he contributed and other books which he edited but we have omitted those. The first of these books,

Notes on Rubik's "Magic Cube" (1980)

Handbook of Cubik Math (1982) with Alexander H Frey

The Cube: The Ultimate Guide to the World's Bestselling Puzzle (2009) with Jerry Slocum, Wei-Hwa Huang, Dieter Gebhardt, Geert Hellings

Problems for Metagrobologists: A Collection of Puzzles With Real Mathematical, Logical or Scientific Content (2016)

Adventures In Recreational Mathematics - Volume I (2021)

Adventures In Recreational Mathematics - Volume II (2021)

*Notes on Rubik's "Magic Cube"*is actually the fifth edition of the notes but the earlier editions had a more limited distribution.**Click on a link below to go to that book**Notes on Rubik's "Magic Cube" (1980)

Handbook of Cubik Math (1982) with Alexander H Frey

The Cube: The Ultimate Guide to the World's Bestselling Puzzle (2009) with Jerry Slocum, Wei-Hwa Huang, Dieter Gebhardt, Geert Hellings

Problems for Metagrobologists: A Collection of Puzzles With Real Mathematical, Logical or Scientific Content (2016)

Adventures In Recreational Mathematics - Volume I (2021)

Adventures In Recreational Mathematics - Volume II (2021)

**1. Notes on Rubik's "Magic Cube" (1980), by David Singmaster.**

**1.1. From the Preface.**

These Notes are a bit like Topsy - they have 'just growed'. There have been so many additions for this edition that I feel it necessary to provide a new and more substantial introduction.

The book has been retitled since the cube is now being sold both as the Magic Cube and Rubik's Cube. This edition is twice the length or the last version of the book and includes a table or contents, the latest news and results, a new notation and diagrams for one face processes, catalogues or useful processes and pretty patterns, many theoretical additions (including the discovery of $PGL(2, 5)$), a bibliography, a detailed index and a detailed step-by-step solution.

The material of the last version has been unchanged except for minor corrections and the liberal insertion of cross references where there was space to add them. These are sometimes abbreviated to just page numbers. I have also inserted a few names of processes or patterns when the names have been bestowed after the original description. There is not always room to give such references, so the reader should remember that a particular topic may be discussed again later and better. The accretion of addenda to the first four editions has been retained as the accretion of new results is felt to reflect the way in which cubism has developed.

The basic material of these Notes is designed to give you, the reader, a basic understanding of the Magic Cube. An algorithm for restoring it to START is developed in section 6 and improved in section 9. A detailed step by step version of it is given separately. With practice, this method takes less than 200 moves and less than 5 minutes. This is quite an accomplishment, but then what? To go beyond, one must have a good notation and some elementary notions of group theory. Section 3 introduces my notation which has been widely accepted and should be readily understandable even to those with no mathematical background. (Indeed, those with mathematical background often adopt much more complex notations.) Section 4 introduces the basic concepts of permutations and group theory, using the symmetries of the square as a standard example. A lot of basic material is quickly covered here and the reader is advised to skim through this section and return to it as necessary. A great deal more group theory develops later. but always as a natural outgrowth of playing with the cube. If the cube fascinates you as much as it does me, you will assimilate an enormous amount of group theory and greatly develop your spatial abilities. It is reported that Rubik actually invented the cube to develop students' three dimensional abilities. The cube is probably the most educational toy ever invented!

There are many exercises and problems, especially in sections 3 to 7. The reader is advised to stop and try them. Some solutions are in section 8, but there are often better answers given later and there are many problems which can be improved or which have not yet been examined. I hope that this presentation is accessible to those with no mathematical background and that it will lead them up to the frontiers of the subject. Though elementary, some of the ideas require practice and hard work to master and the reader is advised to use lots of paper for writing out and for drawing and to constantly have a cube in hand.

Section 1 is an earlier introduction; section 2 is generalities, including how to dismantle the cube. Sections 3 and 4 have been described above. Section 5 introduces the reader to the simpler subgroups of the cube. If your cube is in chaos, get someone to do it for you, or take it apart, or follow the step by step solution, so that you can do this section. These groups are aesthetically pleasing and are small enough that you can easily keep from getting lost. In some of them, you can't even get lost. These explorations should lead you to find sufficient basic processes that you could create a general algorithm.

Section 6 explains and later shows just which patterns of the cube are achievable. It shows how two basic processes can be used to construct an algorithm. This algorithm is greatly improved in section 9. Section 7 introduces a number of further subgroups and problems. Section 8 gives answers to the problems and some comments on the more open-ended problems.

Supplementary section 9 gives many improved processes, including the monoflip and monotwist, leading to a reasonably efficient algorithm. Section 10 presents many new results and extensions. The addenda to the first four editions extend the previous results, giving improvements and many new pretty patterns, discussing the Magic Domino, etc.

Addendum Number Five is nearly as large as all the previous material. It includes the latest anecdotes ('cubist's thumb'), discussion of Thistlethwaite's latest algorithm (at most 52 moves) and other algorithms, a new notation and diagrams for processes that only affect the U face, a small catalogue of useful processes (systematically arranged) and comments on new partial processes (Monoswop, Rubik's Duotwist, Thistlethwaite's Tritwist), a systematic catalogue of pretty patterns and a number of analyses of subgroups (square group, two generator group), among a number of other theoretical results and problems. Some rather advanced topics of group theory turn up on the cube, including $PGL(2, 5)$ in its degree 6 representation and wreath products. A detailed index (5 pages), a bibliography (3 pages), and the step by step solution (4 pages) are included.

Finally, I would like to dedicate this to my wife Deborah, who correctly recognised the cube as an enemy on first sight two years ago and who has wisely refused to touch it, but who has nonetheless gallantly proofread the text.

**1.2. Review by: Dorothy Goldberg.**

*The Mathematics Teacher*

**74**(7) (1981), 576-577.

Rubik's Cube is a cleverly engineered 3 × 3 × 3 cube whose differently coloured faces can rotate about its centre in such a way that the cube remains intact. Just randomly turn the cube several times and the nine unit cubes of each face are transformed into a rainbow of confusion in one of over 3 billion possible combinations. The problem is to restore the scrambled cube to its original state. Singmaster of the Department of Mathematical Sciences and Computing of the Polytechnic of the South Bank in London has produced a timely American edition of his "notes," which could be entitled "Everything You've Ever Wanted to Know about Rubik's Cube but Were Afraid to Ask." You'll be introduced to a system of notation, enough group theory to describe all possible twists and turns, directions for dismantling and maintaining the cube, and an algorithm for restoring the cube to its START position.

**1.3. Review by: Jeffrey Adams.**

*American Scientist*

**71**(2) (1983), 210-211.

Singmaster's book certainly lives up to its billing as "the definitive treatise." Notes started out as an informal set of notes and with the addition of five addenda has grown into a reference work. Every aspect of Rubik's Cube has been touched upon: its mechanics, a notation for moves which has become widely accepted, patterns, a table of useful processes, algorithms, anecdotes. Notes is very mathematical; as far as I know, it is the only book on Rubik's Cube with anything significantly beyond just a solution. How much one gets out of it depends on one's level of interest and mathematical sophistication. Starting with a solution and some easy patterns, the book works through permutation groups, conjugates, and commutators up to topics like wreath products and generation of the cube group. For the millions of cubists who can solve the puzzle and are now looking for something more to do with it, this book is a must.

**1.4. Review by: Mal Davies.**

*Mathematics in School*

**11**(2) (1982), 34.

*Notes on Rubik's Magic Cube*by David Singmaster, one of the world's leading pioneers on the Cube, is your bible on the Cube. It has a four page solution in the centre pages which is concise and can be easily learnt by an able pupil. It also contains a wealth of additional material about the Cube.

**1.5. Review by William M Kantor.**

*Mathematical Reviews*MR0615559

**(82h:20005b)**.

The cube invented by the Hungarian architect and designer E Rubik has produced a worldwide craze. This toy can be purchased on street corners or in toy stores or museums. Many millions have been sold. Competitions have taken place in several countries. On the other hand, Rubik's cube was recently accepted by New York's Museum of Modern Art for its Design Collection.

The author's booklet was written prior to all of this furore. Its purpose is to introduce some of the more elementary group-theoretic aspects of Rubik's cube, to describe an algorithm, and to indicate numerous possible shortcuts, other algorithms and variations on the basic problem of the cube.

The group $G$ of Rubik's cube is the subgroup of $S_{48}$, permuting 48 coloured squares, generated by six specific permutations (clockwise quarter-turns of the front, back, right, left, top and bottom faces of the cube). The cube's mathematical problem amounts to finding an algorithm such that, given $g \in G$, one can determine a sequence $g_{1}, ... , g_{k}$ of powers $g_{i}$ of the generators such that $g = g_{1} ... g_{k}$. This requires the discovery of short words in the generators whose effects on the cube are to move relatively few of the 48 squares. The author describes enormous numbers of such words, and explains how these can be used to write $g$ itself eventually as a word in the generators.

Only at the end of the booklet does he indicate the exact structure of $G$: from the point of view of the booklet, this structure has nothing to do with his algorithm. (In particular, commutators appear somewhat magically throughout his lists of nice words.)

While it is not easy to read, this booklet is very useful as the principal and somewhat encyclopaedic source of mathematical information concerning Rubik's cube.

**1.6. Review by: Rob Evans.**

*New Scientist*(24 September 1981).

*Notes on Rubik's "magic cube"*is rather different from the other books. David Singmaster constructs a solution using basic group theory, which is explained first. The solution is developed later in the book. A mathematical analysis of the cube is given which should be understandable to anyone who is prepared to work through the many exercises. If you do this you will be rewarded with a very good understanding of the cube. Now in its fifth edition, this book contains the results of Singmaster's correspondence with other "cubologists" and gives the latest moves and developments in cube theory. These include such delights as monotwists, U-flips, Cayley Graphs and wreath products (something to do with cube widows?).

**2. Handbook of Cubik Math (1982), by David Singmaster and Alexander H Frey.**

**2.1. From the Publisher.**

The

*Handbook of Cubik Math*unveils the theory involved in Rubik's Cube's solution, the potential applications of that theory to other similar puzzles, and how the cube provides a physical example for many concepts in mathematics where such examples are difficult to find.

The authors have been able to cover and explain these topics in a way which is easily understandable to the layman, suitable for a junior-high-school or high-school course in maths, and appropriate for a college course in modern algebra.

This manual will satisfy the experts' curiosity about the moves that lead to the solution of the cube and will offer a useful supplementary teaching aid to the beginners.

**2.2. From the Preface.**

*Handbook of Cubik Math*is a book about problem solving and some of the fundamental techniques used in problem solving throughout mathematics and science. Both the problems and the illustrations of concepts for solving them are drawn from Rubik's Magic Cube.

Ernö Rubik invented the Magic Cube as an aid to developing three-dimensional skills in his students. Little did he realise the impact that this puzzle would come to have. In 1980 alone, approximately five million cubes were sold. Predictions for future years are that sales will continue at more than twice that rate. In almost every neighbourhood children - and adults - are playing with the cube.

It certainly enhances three dimensional thinking. However, even greater educational value has been found by mathematicians. For them the cube gives a unique physical embodiment of many abstract concepts which otherwise must be presented with only trivial or theoretical examples. Cube processes are non- commutative - that is, changing the order of movements produces different results. Cube processes generate permutations of the pieces of the cube. Sometimes different processes generate the same permutation so that, by looking at the cube, you cannot tell which process was used. This defines an equivalence between processes. The concepts of an identity process, inverses, the cyclic order of a process, commutators, and conjugates all playa part in solving problems on the cube. By experimenting on the cube, a student learns about these concepts and their relation to problem defining and problem solving without having to rely solely on his faith in the teacher or the text.

Perhaps surprisingly, one of the most fundamental concepts which cubik math illustrates is the use of symbolic notation. It is extremely rare to find anyone who can master the complexities of the cube without writing down what movements he has made or is planning to make. Without a good symbolic notation this is cumbersome at best. For communicating about the cube with others a common notation is mandatory.

The

*Handbook of Cubik Math*in the early technical chapters orients the reader to the basic problem of the cube. It introduces a standard notation - one which is internationally accepted. Then it describes a logical method for restoring any scrambled cube to its pristine state where every face is a solid colour. No background of complex or sophisticated mathematical concepts is required in these first three chapters. Many good students in their early teens have mastered these ideas. At the end of Chapter 3, several games are introduced. Playing these will enhance the competitor's understanding of the concepts inherent in controlled modification of the state of the cube.

One might think that after learning how to solve the cube - that is, how to restore it to its monochromatic-sided state - a person would lose interest in the cube. We thought so before we had taught many people how to solve it, only to find that with their increased understanding came increased curiosity. They wanted to understand more about how the cube worked, why processes produced the results they do, and what they could do to enhance their mastery of the cube.

Seldom does one realise at this point that the concepts which appeared so logical for solving the cube problem are, in fact, the concepts of identities, inverses, commutators, and conjugates. Chapter 4 defines these generalised concepts with many examples and exercises from the cube. These principles are applied to derive new techniques for manipulating the cube. Then in Chapter 5 these improvements are applied to obtain better ways to restore the cube.

It is in Chapter 6 that the mathematical concepts become more sophisticated. It is here that the concept of a group is introduced. The structure and the size of the cube group and its subgroups are explored in Chapters 6 and 7. This leads finally to a discussion of normal subgroups and the isomorphisms of subgroups and factor groups in Chapter 8.

It is expected that some students of the cube will only be ready to absorb material through Chapter 3. Others will be able to work through Chapter 5. The more advanced students will work all the way through to the end. At all stages it is necessary to have easy access to a cube. The cube is the best teacher and experimentation is the best learning technique.

**2.3. Review by: Emmett Kinkade.**

*The Mathematics Teacher*

**75**(8) (1982), 709.

I did it! I restored a Rubik Cube by following the directions in the

*Handbook of Cubik Math*. It is heavy reading, but it is readable, the logic is consistent, and the illustrations are helpful. I had successfully resisted the temptations of the cube until I read this book. No more - I now want to know it all. The book is, as advertised, much, much more. It contains games, advanced restoration processes, the mathematics behind the processes, even exercises (with answers). It goes on my list of recommended reading.

**2.4. Review of the 2010 reprint by: David A Reimann.**

*Mathematical Reviews*MR2677319

**(2012d:05085)**.

Rubik's Cube remains one of the most popular puzzles, accessible to nearly all ages and nationalities. This reprinting of the 1982 classic begins with a detailed description of the puzzle, then defines a language and notation for describing the cube's pieces and their movements. This is quickly followed by an algorithmic approach to solving the cube. Beginning in Chapter 4, the reader is gently led into the mathematics of permutations arising in cube movements, including cycles, equivalences, identities, inverses, order, commutativity, and conjugates. Solving the cube is then revisited using these mathematical concepts. The concept of a group is then developed, and the authors show how the permutations of the cube form a group and give examples of certain moves that form cyclic subgroups. The final chapters of the book describe other group theory concepts such as order, permutation parity, nested subgroups, cosets, and isomorphisms. Frey and Singmaster masterfully use the Rubik's cube to give a very tangible and tactile feel to the abstract concept of a group. Many of the sections contain exercises to reinforce concepts presented and to stimulate insight into the mathematics at work in the cube. The solutions in the back of the book promote self-study. The authors conjectured correctly that the smallest number of moves required to solve any scrambled cube was in the low twenties, which was proved to be exactly twenty-two by T Rokicki [

*Mathematical Intelligencer*(2010)]. An updated edition with cube-related mathematics developed in the last 30 years, such as developing the bounds on the number of moves to solve a scrambled cube, would further enhance the book's appeal. Despite the lack of new material, the book remains the definitive source for cube mathematics and presents an interesting and accessible introduction to group theory.

**3. The Cube: The Ultimate Guide to the World's Bestselling Puzzle (2009), by Jerry Slocum, David Singmaster, Wei-Hwa Huang, Dieter Gebhardt, Geert Hellings.**

**3.1. From the Publisher.**

Just in time to satisfy the kids of all ages who received Rubik's Cubes as holiday gifts, here is the book to ensure that the gift won't go unused (or unsolved). Puzzle Masters Jerry Slocum, David Singmaster, Dieter Gebhardt, Wei-Hwa Huang, and Geert Hellings, share their expertise on the greatest puzzle ever created. Not only does

*The Cube*provide dozens of strategies for competitive cubing - including speedcubing - it also shows you how to solve every kind of cube that exists: the 'Mini Cube' 2 × 2 × 2; the 'Classic' 3 × 3 × 3; the 'Rubik's Revenge' 4 × 4 × 4; the 'Professor's Cube' 5 × 5 × 5; and the largest cubes available, the 'V-Cube 6' 6 × 6 × 6 and the 'V-Cube 7' 7 × 7 × 7. The book even includes modular instructions for cubes of any potential size! All solutions are fully illustrated with all-new, easy-to-follow instructions created specially for this book. This unique book also includes a fascinating history of the Rubik's Cube - from its invention in 1974 by Ernö Rubik to the '80s craze to its resurgence today, including dozens of rare photographs of vintage cubes, ephemera, and more! Packed with colour photos from Jerry Slocum's extensive and unique collection of mechanical puzzles,

*The Cube*is a must-have companion to the ultimate brainteaser.

**4. Problems for Metagrobologists: A Collection of Puzzles With Real Mathematical, Logical or Scientific Content (2016), by David Singmaster.**

**4.1. From the Publisher.**

"I believe the book will be welcome by amateur, as well as professional, metagrobologists. Many of the puzzles could be used as warm-up exercises to engender creative atmosphere in a math class. I am sure that many a math teacher will agree with this assessment." Alexander Bogomolny, Cut The Knot.

"Certainly the book is entertaining as it stands, but its real value lies in the opportunity it provides, showing the way with clearly communicated solutions that are examples, but not the whole story." London Mathematical Society.

This book is a collection of over 200 problems that David Singmaster has composed since 1987. Some of the math problems have appeared in his various puzzle columns for BBC Radio and TV, Canadian Broadcasting, Focus (the UK popular science magazine),

*Games and Puzzles*, the

*Los Angeles Times*,

*Micromath*, the

*Puzzle a Day*memo pad and the

*Weekend Telegraph*. While some of these are already classics, many of the puzzles have not been published elsewhere previously.

Puzzle enthusiasts of all ages will find here arithmetic problems, properties of digits; monetary problems; alphametics; Diophantine problems; magic figures; sequence problems; logical problems; geometric problems; physics problems; combinatorial problems; geographic problems; calendar problems; clock problems; dissection problems and verbal problems.

**4.2. Introduction.**

In case you don't already know, the Oxford English Dictionary's (OED) entry for METAGROBOLIZE describes it as humorous. Rabelais used metagroboutizer and Cotgrave (1611) translated it as "to dunce upon, to puzzle. or (too much) beate the braines about". The OED gives: To puzzle, mystify: To puzzle out.

Urquhart's translation of Rabelais used metagrabolizing , mctagrobolism and metagrabolizcd. Kipling used metagrobolized in 1899, which is the latest citation given in the OED.

About 1981, Rick Irby, the American puzzlist, found the verb used in the

*Wall Street Journal*. Since then, the noun has been adopted by a number of puzzlers as a term for one who makes and does puzzles.

Several people have suggested that the noun forms should be metagroboly and metagrobolist but metagrobology is definitely easier to say and this implies using metagrobologist.

As a mathematician and as yet unchristened metagrobologist. I contributed problems to mathematical journals since I was a graduate student in 1963. From 1987, I contributed series of puzzle problems to more popular periodicals and other media - in alphabetical order: BBC Radio 4: BBTV: Canadian Broadcasting: Focus (the UK popular science magazine): Games & Puzzles: Los Angeles Times ; Micromath: The Weekend Telegraph (London). This book is a collection of some of these problems, combined with material contributed to a memo pad - Puzzle a Day (Lagoon Books, London, 2001) - and much previously unpublished material.

My interest in puzzles has always been in understanding, not just the puzzle and its solution, but also the history of the puzzle and its generalisations. All too many puzzle books simply give the specific solution without indicating how to find it. Here I will give reasonably detailed solutions. Some solutions simply require the right insight to yield the answer almost immediately: other problems require setting up algebraic equations and carefully solving them: and some problems require checking a lot of cases and the assistance of a computer is needed. When the problem has some history, I will outline it. Some problems have immediate generalisations which can be stated and solved. In some cases, correspondents have sent me variations of the problem. In a few cases, a generalised problem is unsolved. Word-based puzzles are often rather open-ended. I will be delighted to hear from readers who find simpler solutions or better answers, create other variants of problems, or make progress on the unsolved problems.

People often ask where puzzles come from or how does one find/create puzzles . I must say 'Everywhere'. I am working on a history of recreational mathematics and I read many books containing puzzles. As the reader will see, this directly supplies a number of problems, but such problems are often open to improvement and generalisation because early works give very cryptic and specific solutions. Further, a number of problems arise in everyday life if one is observant - problems about clocks, mirrors, shadows, bicycles, railway trains, letters of the alphabet, etc. Many of these have been examined in the past, but are open to generalisation and variation giving new problems. Another fruitful source of problems is other metagrobologists. There is a pleasure in creating a nice puzzle which makes one want to share it. My friends rightly say that puzzle creators are essentially sadistic, like makers of puns. We want to show a neat problem to fellow connoisseurs, especially if it has a real moment of insight, an 'A-Ha' moment.

The origin of many, perhaps most, classical problems is uncertain. Typically such problems first appear fully developed but with no reference to any earlier versions. Many of these are surprisingly old, deriving from the ancient civilisations, so there is little evidence to elucidate the history. In a few cases, one can see a distinctive problem appearing in China, then India, then in the Arabic world and then in medieval Europe - but we know nothing about how the problem was transmitted along this route. In other cases, we only see the problem in two distant places, e.g. at the far ends of the Silk Road. Very rarely, we can see a problem is new at a particular time because people are having difficulty with it and getting it wrong. But even modern problems can be hard to trace - I have several problems that arose in living memory but the originators cannot be determined. Mathematical problems are like urban legends or jokes - they are part of the folklore of mathematics.

I am grateful to many collectors and creators of mathematical puzzles in the past, whose works are regularly pirated here and elsewhere, just as they pirated from their predecessors, but I will try to acknowledge them and to give new versions of their problems. Of these, the most famous are Sam Loyd, Henry Dudeney, Hubert Phillips and Martin Gardner. Gardner is by far the most important expositor of mathematical recreations of all time - he was still writing at the age of 90. He has also created some good puzzles.

Unfortunately, when I wrote some of these problems, I simply referred to, e.g. a 1930s puzzle book, and I have not [yet] relocated the source. As will be seen, puzzles occur in diverse places.

But enough waffle. Let us get down to puzzles.

**4.3. Review by: Peter Winkler.**

*The American Mathematical Monthly*

**124**(8) (2017), 763-768.

Singmaster's ... 221 puzzles span many topics, and no general approach to solution is attempted. Instead, the author intrigues his audience with connections to "real" problems, together with historical tidbits. The jacket claims the author "composed" these puzzles, but - and this is a good thing for the reader - many are classics, dug up from old books where they were misstated or incorrectly solved, to which Singmaster has given new life (often by asking for all solutions to a problem that normally asks for only one).

For example, Puzzle #4 concerns sequences of $n$ numbers whose sum and product are the same. It turns out this problem is solvable for any $n$ but uniquely solvable only, as far as anyone knows, for a few values of $n$ of which the largest is 444. (The author tested up to $n = 7550$, but others have looked in vain up to ten million or more.)

History adds interest to many of Singmaster's conundrums. Puzzle #151 shows a photo of an actual gravestone commemorating someone who seems to have died first and been born later; wouldn't you like to hear the explanation? Of course you would! Many others ask good questions to which you might reasonably want to find answers, even when the answer might be disappointing.

But perhaps the greatest benefit to be enjoyed by the metagrobologists (puzzle mavens) to whom Singmaster's book is addressed is the reminder that puzzles are living things. Even if you have seen many of these puzzles before - nay, especially if you have seen them before - a reacquaintance is well worth the time spent.

**5. Adventures In Recreational Mathematics - Volume I (2021), by David Singmaster.**

**5.1. From the Publisher.**

David Singmaster believes in the presentation and teaching of mathematics as recreation. When the Rubik's Cube took off in 1978, based on thinly disguised mathematics, he became seriously interested in mathematical puzzles which would provide mental stimulation for students and professional mathematicians. He has not only published the standard mathematical solution for the Rubik's cube still in use today, but he has also become the de facto scribe and noted chronicler of the recreational mathematics puzzles themselves.

Dr Singmaster is also an ongoing lecturer of recreational mathematics around the globe, a noted mechanical puzzle collector, owner of thousands of books related to recreational mathematical puzzles and the "go to" source for the history of individual mathematical puzzles.

This set of two books provides readers with an adventure into previously unknown origins of ancient puzzles, which could be traced back to their Medieval, Chinese, Arabic and Indian sources. The puzzles are fully described, many with illustrations, adding interest to their history and relevance to contemporary mathematical concepts. These are musings of a respected historian of recreational mathematics.

**5.2. From the Preface.**

I have always been interested in various mathematical recreations. But this interest became serious after the Rubik's Cube craze in 1978 and onward. The public showed an unprecedented interest in an activity that was only thinly disguised mathematics. I was very much in the midst of this furore when I wrote the first book on the Cube and how to solve it.

As a mathematician, I was aware of the principles the Cube displayed: the symmetries, the power of a sequence of moves to switch pieces of the Cube while not moving most of the rest, called conjugates. Mathematicians had explored the principles a century before; they called it Group Theory. This opened a door wide for me. I began to revisit the many topics in recreational mathematics and pursued their origins. When did the basic puzzles or problems first appear? Who introduced them first? How did they change over time? And how might they inspire today's enthusiasts to take them further? I thought it would be possible to produce a book on the origins of recreational problems.

When I embarked on this, I soon discovered that such information did not exist - for many problems, the origins were completely unknown. I then read numerous works and uncovered early origins of problems in Latin works and in translations of Chinese and Indian works. I began to make notes on early sources. This has grown to a document of about 1000 pages covering 472 topics. I say more about this in the Appendix .

In this process, I have discovered and published material on the origins of numerous problems in various publications. This book is my attempt to gather these results together in one place, in a standard form with many updates. These collective efforts are my own "adventures in recreational mathematics" .

Some examples of my adventures are as follows :

• Chapter 4 studies Alcuin, who wrote one of the earliest European mathematical texts, c. 800; it was explicitly a text with recreational exercises. This is not well known. So I undertook to explore the prior scholarship of the Latin text and to annotate Alcuin so the modem reader can easily understand it.

• Chapter 5 studies the Latin text of Abbot Albert, c. 1150; it includes puzzles. I have translated these portions for the modern public, incorporating prior scholarship.

• Chapter 6 presents the work of Pacioli, c. 1500, who published the first book on recreational mathematics. He introduced so many puzzle types that every interested student should know him. However, his output remains hidden in scholarly works. I worked through the various translations and competing ideas to assemble a cogent presentation on Pacioli.

Therefore, much of this volume is not reporting on others' work, but presenting new analyses, translations, and categorizations.

Another aspect of this book is found in Part I1, where I have taken up ancient puzzles and found them to be sources of open questions. This explores the "mathematics" of recreational mathematics. Using today's mathematical toolbox, we can ask and answer questions the old masters could only hint at.

A majority of the sources are manuscripts and other documents that are difficult to find and to cite. As a remit, the last chapter gives detailed information about these sources, since including it after each chapter would be both lengthy and repetitive. So if a chapter's bibliography has a reference in a font like this, PACIOLI-DVQ, then it indicates that the full citation is in the Appendix.

The companion volume to this is 'New Adventures in Recreational Mathematics'. Whereas the emphasis in this book is on the historical roots of these puzzles and problems, the second book mainly concentrates on recently posed problems. In that book not only is the presentation more modern, but also the mathematics is slightly more difficult.

Dedicated to the memory of great puzzlers and friends: Martin Gardner, John Conway, Solomon Golomb, Richard Guy, Edward Hordern and Nob Yoshjgahara. My thanks go to Paul J Campbell for suggestions and to Mike Hammerstone for technical support.

**5.3. About Singmaster.**

David Breyer Singmaster studied at Caltech and received a Ph.D. in mathematics from the University of California-Berkeley in 1966. He taught at the American University of Beirut, later lived in Cyprus, and then came to London in 1970 - and has been based there since. He retired from London South Bank University in 1996 and was designated emeritus in 2020. His interests are in number theory and combinatorics, and the history of mathematics and of science in general.

From 1978 to about 1984, he was the leading expositor of Rubik's Cube. He devised the non-standard notation for it, wrote the first book on the Cube, and later edited Rubik's book into English. Due to revived interest in the Cube, he and some colleagues produced a new book

*The Cube: The Ultimate Guide to the World's Bestselling Puzzle - Secrets, Stories, Solutions*in 2009 and

*The Handbook of Cubik Math*in 2010.

Since about 1982, be has been working on a history of recreational mathematics,

*Sources in Recreational Mathematics*, see Appendix A, which has involved reading and studying mathematics from every culture and period.

He was the opening speaker at the Conference opening the Strens Memorial Collection at Calgary in 1986. He has attended all of the Gatherings for Gardner and spoken at many of the them. He acted as Chairman for four of these. He has attended all of the MathsJam's in England. He was an invited speaker at the Third Iberian Colloquium on Recreational Mathematics in 2012, where he spoke on "Vanishing Area Puzzles" (Chapter 16). He attended the Fourth Colloquium in Lisbon in 2015, where he spoke on "Early Topological Puzzles" (Chapter 8). His book

*Problems for Metagrobologists: A Collection of Puzzles With Real Mathematical, Logical or Scientific Content*, a collection of over 200 problems that he composed since 1988, appeared in 2015.

**6. Adventures In Recreational Mathematics - Volume II (2021), by David Singmaster.**

**6.1. From the Preface.**

My first published paper was "On round pegs in square holes and square pegs in round holes" (1964), reproduced as Chapter 2. Martin Gardner mentioned the result in his Scientific American column. This encouraged me to not only persist with this problem, but to engage in other recreational topics.

This interest became serious after the Rubik's Cube craze in 1978 and onward. I then thought it would be possible to produce a book giving the origins of recreational problems. When I embarked on this, I soon discovered that this information had never been collected and was largely unknown. This led to decades of research into the origins of these puzzles and problem. This is the subject of the companion volume

*Adventures in Recreational Mathematics*.

However, even fairly recent puzzles have obscure origins. For example, the famous problem where one has 12 coins, one of which is counterfeit and weighs more or less than the others and one has to find which coin is counterfeit and whether it is heavy or light in three weighings appears to have evolved within living memory, during World War II, but no one claims to have invented it. This volume is devoted to recently posed problems.

The opening chapter explains the breadth of the topic, and I use a wide definition. Others might use a narrower scope. When dealing with recently posed problems every mathematical discussion uses mathematical tools. Those tools have a history and that history may lead back to a recreational topic. In this volume, we usually do not follow such threads back in time, focusing instead on the new problem.

The mathematical sophistication varies between chapters. Some of the articles collected in this volume appeared in undergraduate mathematics journals. There is no advanced mathematics herein, but sometimes I assume the reader is a beginning undergraduate. In general, this volume requires more knowledge than what was required for the companion volume. This means if you only have mastered high school algebra and geometry there may be some passages that you will want to skim over. When prior experience is lacking, you are only as ked to accept the prior knowledge quoted. Hopefully any passages that are not clear on first reading will encourage further study.

The first chapter is more discursive, discussing the utility of recreational mathematics. In doing so it briefly goes over some historical facts covered in more detail in the companion volume. In each subsequent chapter I pose a problem. In some cases it continues some recent research by others. In other cases it is a problem that I may be the first to pose. However, generally, the rest of this book is a series of chapters motivated by curiosity.

Mathematics is the "queen of the sciences". And like science it is a combination of curiosity-based research and settling old conjectures. This book is mainly curiosity-based. What makes it "recreational" is the very fact that the problems excite our interest without the need for some application to motivate them. We have fun solving them because we are curious what the answer is. This is in contrast to restricting ourselves to "puzzles".

For example, I studied the Sum = Product sequences for some time in the late 1980s. This concerns sets of numbers, like {1, 2, 3} which sum to the same number as their product. There is no need for an application to pursue this topic, we just want to know. Further, any setter of puzzles worth his or her salt can take the results (in Chapter 4) and make a puzzle out of them.

The final chapter is more of a coda. It takes a recreation of uncertain age, which is more of an optical illusion than a puzzle, and shows how widespread it is today. I hope you find these problems as entertaining as I found them when solving them.

Last Updated September 2023