The fascination with mechanical puzzles and recreational mathematics is as old as human civilisation. Since time immemorial, problems have been presented in fanciful settings to make mathematical concepts more easily intelligible. The first book to use recreational mathematics in the title dates from 1624. It was published at Pont-à-Mousson in Alsace, with no indication of authorship on the title-page. Only three copies of the first edition are known to exist. The dedication is signed by Henrik van Etten, a student at the university of Pont-à-Mousson, where Jean Leurechon taught mathematics, and to whom this work is traditionally attributed. In 1954 the Compagnie de Pont-à-Mousson, a French steel manufacturer, produced a handsome facsimile of the first edition, which is itself a rarity.
It was the Rubik’s Cube craze that first inspired me to write a book on the origins of recreational mathematics. Readers may be familiar with the classic river-crossing and bear-hunting problems without realising that they can be traced back to antiquity. After half a century of research, during which I accumulated 12,000 books, 3,000 puzzles and one cuneiform tablet, my two-volume Adventures in Recreational Mathematics was published in 2021.
Although I’m frequently accused of being English, I was born in Missouri and educated at Caltech and UC Berkeley. I moved to London in 1970 and embarked on a teaching career in mathematics at London South Bank University. I had picked up a certain number of books on the history of mathematics as a graduate student, but it wasn’t until I came to London that I started accumulating them in quantity. I would go to the monthly book fairs, where the dealers knew what I was looking for, and would put interesting items aside. Eric Korn found me a copy of The Tangram Book, 1817, the oldest work in English on the subject, which started a craze for the Chinese puzzle.
Children’s books have always been a good source of recreational mathematics. Unfortunately they often command high prices if they are in fine condition, whereas I’m only interested in their content. Newspapers and magazines are another good source. Henry Dudeney, England’s greatest creator of mathematical puzzles, began his career by contributing problems to weekly publications, including The Weekly Dispatch. I have an album in which the original owner cut out Dudeney’s puzzles from The Weekly Dispatch, and kept them with his workings, and a note recording if he won the prize for that week. Dudeney’s first book, The Canterbury Puzzles was published in 1907.
One of the first bookshops that I visited regularly was Mr Linden’s in Craven Street, off the Strand. He had four interconnected basements, full of books on the history of science quite haphazardly organised. It wasn’t easy to move around because the aisles were piled high with books. When I suggested to Mr Linden that he might sell more books if he organised them better, he replied, ‘Ah, but it’s very hard to replace them’. One of the piles contained the remnants of the library of G.H.Hardy. Although most of his books had been given to the London Mathematical Society, others turned up here and there. A friend found some on a barrow in a local market and, through him, I managed to acquire a book with Hardy’s signature. I also bought a copy of John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior with Morgenstern’s signature.
In 1971 I won a research scholarship at the Istituto di Matematica in Pisa, and my wife and I spent a year in Italy, a country that we both love. In Italy there is a law that requires banks to spend 2% of their profits on ‘good works’, which can be interpreted as publishing facsimiles of medieval manuscripts on mathematics. A few years before we arrived in Tuscany, the Cassa di Risparmio di Firenze published a facsimile of the magnificent illuminated manuscript of Filippo Calandri’s Trattato di Arithmetica, Florence, 1491. The manuscript is one of the treasures of the Biblioteca Riccardiana in Florence. It contains 230 miniatures in gold and silver, some attributed to the workshop of Botticelli, and many depicting recreational problems set against Italian cityscapes. The manuscript was commissioned by Lorenzo the Magnificent for the education of his son, Giuliano de’ Medici, the future Pope Leo X. When Italian banks sponsor facsimiles and reprints, they tend to do so in limited numbers for presentation to visiting dignitaries and business associates. It took me thirty years to find my own copy of the Calandri facsimile, which finally turned up in a Florentine bookseller’s catalogue.
Back in London I returned to my teaching career and, in 1978, I got my hands on my first Rubik’s Cube. I was at the International Congress of Mathematicians in Helsinki, where Trevor Fletcher, HM Inspector of Mathematics, managed to get hold of a cube. Incidentally, Trevor’s wife was one of the first people to suffer from Cubist’s Thumb. Trevor was able to get a cube for me from Professor Tamás Varga, the distinguished Hungarian mathematician who had brought some of the first cubes out of Hungary. It took me two weeks - on and off - to find a solution, and in October 1979, I self-published Notes on the “Magic Cube”, containing my now standard notation for solving the cube. An expanded fifth edition, retitled Notes on Rubik’s “Magic Cube” was published in August 1980, and included the results of my correspondence with other Cubologists. If readers solved the cube using the Singmaster notation, I suggested in the book that they scream hooray, buy a round of drinks, send me a cheque, and tell the orderlies that they could be let out now.
Ernö Rubik remarked that the beauty of my notation was that it was based on the human body – right, left, up, down, front, back – and that he always tried in his designs to incorporate the human aspect. At the height of the craze in the 1980s, the cube was without doubt the outstanding puzzle of the century. There were more cubes sold in Hungary than there were people. I have around 400 cubes and their variants, including a lot of illustrated cubes, and more keep appearing - there’s even a Sudokube. A version of Sudoku appeared in the writings of Jacques Ozanam, whose Récréations Mathématiques et Physiques, Paris, 1694, contains a problem in which the reader has to arrange the sixteen court cards so that each row and each column contains one of each suit and one of each value.
I’ve met Rubik on several occasions, most recently at a G4G Conference in Atlanta. The first conference was held in 1993, and I’ve attended all the biennial gatherings, and spoken at many of them. G4G stands for ‘Gatherings for Gardner’, and is a tribute to the life and work of Martin Gardner, who wrote the legendary ‘Mathematical Games’ column in Scientific American. Although Gardner had no formal training in mathematics, his column, which first appeared in January 1957, was renowned for presenting mathematical ideas in engaging and non-technical ways. It was a major factor in the popularity of the journal, and probably inspired more students to study mathematics than any other influence. Gardner was a lifelong admirer of the works of Lewis Carroll, with whom he shared a love of numerical riddles. His Annotated Alice waspublished in 1960, and sold over a million copies.
G4G was established in Atlanta by the late Tom Rodgers, a great puzzler, who is probably better known in the book world as a collector of dictionaries. Rodgers came from an old Georgia family, who put up the money for Jimmy Carter to run for Governor. He lived in a magnificent house in Atlanta, where he would give a picnic during conferences. Tom Rodgers died in 2012, two years after Martin Gardner. The Gatherings continue in Gardner’s honour, and remain true to Rodgers’ vision of bringing together mathematicians, puzzle enthusiasts and magicians.
I was one of the contributors to A Lifetime of Puzzles, a festschrift for Martin Gardner, published in 2008. My subject was the first recreational mathematics book, written by Luca Pacioli, the friend and collaborator of Leonardo Da Vinci. De Viribus Quantitatis (On the Powers of Numbers) is regarded as the foundation of modern magic and numerical puzzles, and is believed to be the first documentation of magic tricks with cards and coins. There is quite an overlap between people who are interested in the history of magic, and those who specialise in recreational mathematics. Pacioli’s book is an early example of this overlapping interest, which can be found in many later books on puzzles, particularly if the magic depends on binary arithmetic. The Harry Price Library of Magical Literature in Senate House London is a wonderful collection of nearly 13,000 books, pamphlets and periodicals on all aspects of magic from legerdemain to psychical research. Harry Price was active in the early twentieth century as an exposer of paranormal fraud. He was a forerunner of Houdini, who was also determined to expose the fraudulent ways of Spiritualist mediums. One of the most important collectors of magical literature is Bill Kalush, whose biography of Houdini was published with Larry Sloman in 2006. Kalush’s library forms the basis of his Conjuring Arts Research Center in New York. I was on the board of directors for a while with David Blaine, the magician, who works closely with Kalush.
I first saw Pacioli’s De Viribus Quantitatis on an incomplete microfilm at the Warburg Institute. The film ended tantalisingly at the start of an interesting section. Fortunately one of my fellow puzzlers lives in Bologna, the location of the original manuscript, and was able to photograph it and send it to me on a CD. The manuscript was written between 1496 and 1508, and has been in the Biblioteca Universitaria since the eighteenth century, having previously been in private collections. A nineteenth-century manuscript copy exists in the Biblioteca Casanatense in Rome. In 2007 an article appeared in The Guardian entitled, ‘And that’s renaissance magic: Compendium of card tricks, number puzzles and illusions written by Leonardo Da Vinci’s best friend’. It claimed that I had rediscovered the manuscript. Actually I had simply discovered that it contained much more interesting and important material than had been realised.
However I can claim responsibility for the first complete English translation of the Propositiones ad acuendos juvenes (Problems to Sharpen the Young), attributed to Alcuin of York, Charlemagne’s Minister of Education. John Hadley, a Catholic priest and Latinist, kindly translated Alcuin’s text for me, and we co-published an article on ‘Problems to Sharpen the Young’ in The Mathematical Gazette, 1992. One of the sources for the Alcuin text is Jacques-Paul Migne’s Patrologiae Latinae, from which John Hadley made his English translation, and I annotated this with my knowledge of the various problems, their origin and first appearance. Migne’s monumental anthology was published in the 1860s, and contains the works of the Church Fathers from Tertullian to the death of Pope Innocent III in 1216. In a letter to Charlemagne, Alcuin mentions sending to the Emperor ‘certain subtle figures of arithmetic, for pleasure’.
Although there is nothing definite to connect this letter with the Propositiones, Migne included them among the doubtful works of Alcuin. In an editorial comment, Migne notes that, although the Propositiones can easily be solved by modern arithmetic, ‘there were those who, attentive to certain relics of antiquity, would find a value in them’. And indeed the collection of 56 problems includes seven major types of problem which appear for the first time, and two major types which appear for the first time in the West. The number of topics that have Oriental origins emphasises our increasing realisation that modern algebra and arithmetic derive more from Babylonia, China, India, and the Arabs than from Greece. I have a cast of an Old Babylonian tablet, ca.1800 BC, listing Pythagorean triples. The original tablet, known as Plimpton 322, is in Columbia University. I persuaded them to make casts from the original, and I believe copies may still be available from their Rare Book Room.
A major question is – where did Alcuin find his problems? No doubt he was able to make use of the network of cultural contacts within the Carolingian Empire which, at its greatest extent, had a border with the Emirate of Cordoba. A number of the problems are identical to those known in the Arabic tradition. Alcuin recorded for the first time the camel-carrying-provisions problem, which inspired a solution for refuelling aircraft during the Second World War, and led to the development in the 1950s of dynamic programming, a mathematical optimisation method. It’s a good example of recreational mathematics’ ability to inspire new ways of thinking. Solving problems naturally develops problem-solving techniques, Other problems in Alcuin have roots going back to the Ancient Near East, and the ‘100 Fowls’ problem is derived from the work of a fifth-century Chinese mathematician Zhang Qiujian: A man buys 100 fowls for 100 cash (an old coin). Roosters cost 5, hens 3, and chicks are 3 for a cash – how many of each did he buy? Answers on a postcard please.
The question for me at the moment is what to do with my collection. It isn’t easy to find somewhere that will accept objects as well as books. Storing mechanical puzzles can be quite a challenge. Martin Gardner kept his collection in 29 filing cabinets and a disused fridge. Jerry Slocum found a home for his mechanical puzzles at the Lilly Library, Indiana, where they were willing to accept over 30,000 puzzles as well as books. Jerry provided for a purpose-built space, and the appointment of the world’s first puzzle curator. As my library is basically an English and European collection, I think it ought to stay in England. The Mathematical Association has a library at Leicester University, but their catalogue is incompatible with COPAC. Ideally I would like my books to be easily located online so that people know where they can find them. There really is considerable interest in mathematics out there and, if we enjoy our subject, it should be our duty and our pleasure to try to encourage and feed this interest.
Interviewed for The Book Collector Summer 2022
David Singmaster died on 13 February 2023