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David Singmaster

David Singmaster (14 December 1938 – 13 February 2023) was an American-British mathematician best known for his pioneering work in recreational mathematics, puzzle theory, and the mathematical analysis of the Rubik's Cube.^[1]^[2] Born in Ferguson, St. Louis, Missouri, Singmaster grew up in Bainbridge, New York, the son of James Arthur Singmaster and Marjorie Lowe Breyer.^[1] He initially studied civil engineering and then organic chemistry at the California Institute of Technology from 1956 to 1959 but was expelled after his third year.^[1] Undeterred, he transferred to the University of California, Berkeley, where he earned a Ph.D. in mathematics in 1966 under Derrick Henry Lehmer, with a thesis titled "On means of differences of consecutive integers relatively prime to m."^[1] His early career included teaching positions in Beirut, Lebanon, and Pisa, Italy, as well as a stint as a photographer on an underwater archaeological expedition off Sicily, where his team discovered the oldest known warship wreck.^[2] In 1970, Singmaster settled in London and joined the Polytechnic of the South Bank (later London South Bank University) as a lecturer in mathematics, rising to professor in 1992 and readership in 1984.^[1]^[3] He retired early in 1996 but continued as an honorary research fellow at University College London and emeritus professor at London South Bank University until 2020.^[1]^[3] Throughout his career, Singmaster was deeply engaged with the British Society for the History of Mathematics, organizing events, delivering over a dozen talks on topics like the Rubik's Cube and historical puzzles, and creating the "Mathematical Gazetteer of the United Kingdom," a comprehensive catalog of mathematical sites serialized from 1995 to 2003.^[3] Singmaster's most enduring contributions lie in recreational mathematics and puzzles, where he amassed a collection of over 3,000 mechanical puzzles, including 400 Rubik's Cube variants, and 12,000 books on the subject.^[4] He solved the Rubik's Cube shortly after first encountering it in 1978, devising the standard Singmaster notation for describing moves and publishing the first English-language solving guide, Notes on Rubik's "Magic Cube" (1979).^[2]^[3] He edited the influential newsletter Cubic Circular from 1981 to 1985, fostering the cubing community and exploring its group theory.^[3] Other notable works include his translation of the medieval puzzle collection Propositiones ad acuendos juvenes (1992), the rediscovery of Luca Pacioli's De Viribus Quantitatis (2007), and late-career books like Problems for Metagrobologists (2016) and Adventures in Recreational Mathematics (2021), which compile historical puzzles and his own inventions.^[1]^[4] He also proposed Singmaster's conjecture in 1982, positing a finite upper bound on the frequency of entries in Pascal's triangle (with 3003 appearing eight times as the known maximum).^[2] With over 60 publications cataloged in MathSciNet, Singmaster's enthusiasm for sharing puzzles at events like Gathering for Gardner influenced generations of mathematicians, including friendships with John Conway, Richard Guy, and Martin Gardner.^[3]^[2]

Early life and education

Family and childhood

David Singmaster was born on December 14, 1938, in Ferguson, a suburb of St. Louis, Missouri, USA.^[1] He was the second of four children born to James Arthur Singmaster and Marjorie Lowe Breyer Singmaster. His father, born in 1908 in Palmerton, Pennsylvania, held a B.A. from Williams College (1929) and an M.B.A. from Harvard University (1931); he worked as a chemist at Mallinckrodt Chemical Works in St. Louis from 1931 to 1941 before managing the Borden Chemical Plant in Bainbridge, New York, starting in 1941, and later joining Monsanto Chemical Company, retiring as a marketing executive. His mother, born in 1910 also in Palmerton, graduated from Wellesley College in 1932 as a Phi Beta Kappa member.^[1]^[5] The couple had married on November 28, 1931, in Bronxville, New York, and their other children were James Arthur III (born circa 1935), Allan (born circa 1942), and Karen (born circa 1949).^[1] In 1941, the family relocated to Bainbridge, New York, due to his father's new position, where Singmaster spent the remainder of his childhood. He attended local schools in Bainbridge, including high school, where he demonstrated an early aptitude for numerical tasks by serving as Assistant Treasurer of the General Organization and managing a $25,000 budget.^[1] This upbringing in a household shaped by his parents' scientific and educational backgrounds laid the foundation for his later pursuits in mathematics.^[1]

University studies

Singmaster began his undergraduate studies at the California Institute of Technology (Caltech) in 1956, initially pursuing civil engineering before switching to organic chemistry.^[1] Facing academic challenges, including an expulsion at the end of his third year in 1959 due to lack of academic ability and lock-picking activities, he left Caltech without a degree and spent a year working before transferring to the University of California, Berkeley, in 1960, where he majored in physics while taking extensive mathematics courses.^[1] He completed his Bachelor of Arts degree in physics at Berkeley around 1961.^[1] Following his undergraduate graduation, Singmaster continued directly into graduate studies at the University of California, Berkeley, supported by a National Science Foundation Graduate Fellowship.^[1] His early research during this period focused on number theory, particularly sums of consecutive integers and their connections to odd factors, leading to his first publication in 1964 on geometric fitting problems generalized to n dimensions.^[1] Under the supervision of D. H. Lehmer, a prominent number theorist, and with additional guidance from R. Sherman Lehman, Singmaster completed his PhD in mathematics in 1966.^[3]^[1] His dissertation, titled On means of differences of consecutive integers relatively prime to m, explored arithmetic properties related to sequences of integers coprime to a modulus m, emphasizing probabilistic and analytic aspects without delving into advanced derivations.^[1] This work laid the groundwork for his ongoing interests in analytic number theory.^[1]

Academic career

Positions in the United States

Singmaster began his academic career in the United States with graduate studies at the University of California, Berkeley, where he earned his PhD in mathematics in 1966 under the supervision of Derrick Henry Lehmer.^[1] His doctoral thesis, titled On means of differences of consecutive integers relatively prime to m, explored analytic aspects of number theory, focusing on averages of differences between pairs of consecutive integers coprime to a fixed modulus m; this 59-page work was supported by a National Science Foundation Graduate Fellowship.^[1] During his time at Berkeley from 1960 to 1966, Singmaster engaged in research in number theory and published several influential papers that built on his thesis interests. Notable among these was "A Maximal Generalization of Fermat's Theorem" (1966), which extended Fermat's Little Theorem by considering maximal sets of residues modulo p where certain congruence conditions hold, providing new insights into modular arithmetic and prime power behaviors.^[1] Another early contribution, co-authored with his wife Geralda Singmaster, was "Forbidden Regions Are Convex" (1967), addressing geometric constraints in optimization problems, though published shortly after his PhD.^[1] These works established Singmaster's early reputation in pure mathematics, emphasizing rigorous proofs in arithmetic progressions and convex sets. Singmaster's foundational training at Berkeley prepared him for subsequent international roles, but his direct academic positions in the US concluded with his doctorate.^[1]

Career at London South Bank University

Following his graduate studies in the United States, a brief teaching position at the American University of Beirut in 1966–1967, and a period of residence in Cyprus, David Singmaster relocated to the United Kingdom in 1970 to pursue a lecturing position at the Polytechnic of the South Bank in London.^[1]^[3] He joined the Polytechnic of the South Bank as a lecturer in the Department of Mathematics in 1970.^[1] The institution was renamed London South Bank University in 1992, under which Singmaster continued his career as Professor of Mathematics in the School of Computing, Information Systems and Mathematics.^[3] He was promoted to Readership—a research professorship—in September 1984, becoming the first in his faculty to hold this title.^[1] In 1972–1973, Singmaster held a fellowship from the Italian National Research Council at the Istituto Matematico in Pisa, Italy.^[1] He also had a visiting position at City College of New York in autumn 1985.^[1] Singmaster formally retired from teaching in 1996 but maintained an affiliation with the university and was elected Professor Emeritus in 2020.^[1]^[3] Throughout his tenure, Singmaster taught core mathematics courses, including combinatorics and number theory, contributing to the curriculum in applied and theoretical areas.^[3] In the 1983–84 academic year, he voluntarily taught additional courses beyond his standard load, introducing a new geometry course to expand student offerings.^[1] These efforts helped enhance the department's teaching portfolio and engaged students with diverse mathematical topics. Singmaster also made institutional contributions by organizing events to promote mathematical interests, such as a 1996 meeting on the history of recreational mathematics held at the university.^[3] Around the late 1970s, his research and teaching focus shifted from pure number theory toward integrating recreational mathematics and puzzles, reflecting a broader evolution in his professional interests while at the institution.^[1]

Mathematical contributions

Research in number theory

Singmaster's research in number theory primarily focused on combinatorial aspects, particularly the arithmetic properties of binomial coefficients and related structures. During his doctoral studies at the University of California, Berkeley, where he earned his PhD in 1966 under advisor D. H. Lehmer, Singmaster investigated the mean differences of consecutive integers relatively prime to a fixed modulus m. This work examined the average spacing between such integers, providing insights into their distribution and contributing to analytic number theory concerns about coprimality patterns.^[6] Following his time in the United States, including teaching positions after his PhD, Singmaster continued this line of inquiry in the United Kingdom at the Polytechnic of the South Bank (now London South Bank University). He published several papers on the divisibility properties of binomial coefficients. For instance, in a 1974 paper, he determined the smallest n such that a prime power p^e divides the binomial coefficient \binom{n}{k} for some k, establishing explicit bounds and conditions based on Lucas' theorem and Kummer's generalization. This result advanced the understanding of prime power divisibility in Pascal's triangle entries. In a companion paper that year, Singmaster proved that for any fixed integer d > 1, the set of binomial coefficients divisible by d has natural density 1 among all binomial coefficients, extending earlier results on prime divisibility to composite integers and highlighting the ubiquity of such divisibility.^[7] Singmaster also contributed to the theory of polynomial functions modulo m. In another 1974 publication, he derived canonical representations for integer polynomials that vanish modulo m and for the residue classes they induce, using concepts from finite differences and interpolation. These representations facilitate the study of polynomial behaviors over modular rings, with applications to enumerative problems in combinatorics.^[8] Additionally, he explored divisibility criteria for both binomial and multinomial coefficients by primes and prime powers, providing theorems on the exact powers of primes dividing these coefficients under various conditions, which built on classical results like those of Glaisher and von Szily.^[9] These contributions, appearing in journals such as the Journal of Number Theory and the Journal of the London Mathematical Society, emphasized conceptual properties like divisibility and modular vanishing rather than exhaustive computations. Singmaster's work on binomial coefficients influenced later developments in combinatorial number theory and found applications in recreational mathematics, such as analyzing numerical patterns and symmetries in triangular arrays without focusing on repetition frequencies.^[1]

Singmaster's conjecture

Singmaster's conjecture posits that no positive integer greater than 1 appears more than eight times as an entry in Pascal's triangle, meaning the multiplicity of any such binomial coefficient \binom{[n](/page/N+)}{[k](/page/K)} is at most eight.^[10] This bounded multiplicity implies that for any integer m > [1](/page/1), the equation \binom{[n](/page/N+)}{[k](/page/K)} = m has at most eight solutions in nonnegative integers n \geq k \geq 0.^[11] The conjecture specifically targets the finite repetition of values beyond the ubiquitous 1, which appears infinitely often along the triangle's edges.^[12] Proposed by David Singmaster in 1971, the conjecture arose from computational explorations of repeated entries in Pascal's triangle, such as the number 6 appearing three times as \binom{4}{2}, \binom{6}{1}, and \binom{6}{5}.^[13] Singmaster's initial work established that the multiplicity N(m) satisfies N(m) = O(\log m), providing an upper bound that grows slowly but suggesting a possible absolute cap based on observed patterns up to modest row sizes.^[14] These observations highlighted how small composites like 6 recur modestly, contrasting with 1's infinite presence and motivating the search for a universal limit on repetitions.^[10] Extensive computational verification supports the conjecture, with no counterexamples found despite checks extending to rows beyond n = 10^6.^[15] The highest confirmed multiplicity is eight, uniquely realized by 3003, which appears as \binom{14}{6}, \binom{14}{8}, \binom{15}{5}, \binom{15}{10}, \binom{78}{2}, \binom{78}{76}, \binom{3003}{1}, and \binom{3003}{3002}.^[16] No integers are known to appear exactly five or seven times, though multiplicities of six are observed for several numbers, such as 3003 also exemplifying higher repetitions in related contexts.^[14] These results underscore the rarity of high multiplicities, with ongoing efforts focusing on asymptotic bounds in the triangle's interior.^[17] The conjecture connects to deeper number-theoretic principles, particularly through prime factorizations of binomial coefficients and Kummer's theorem, which counts the exponent of a prime p in \binom{n}{k} as the number of carries when adding k and n-k in base p.^[12] This p-adic valuation framework limits how the prime factors of m can align across multiple \binom{n}{k} = m, as each solution requires compatible carry patterns without exceeding the fixed factorization of m.^[15] Such connections have enabled partial progress, like exponential sum techniques bounding multiplicities away from the triangle's boundaries, though the full conjecture remains open.^[17]

Recreational mathematics and puzzles

Development of puzzle interests

David Singmaster's interest in puzzles emerged during his graduate studies in the United States in the 1960s, particularly at the University of California, Berkeley, where he was pursuing his Ph.D. in mathematics. As a student, he began collecting books on the history of mathematics, influenced by a housemate's work on polyomino puzzles and the recreational aspects encountered in number theory courses.^[1] This period marked the inception of his engagement with recreational mathematics, sparked further by Martin Gardner's "Mathematical Games" column in Scientific American, which debuted in 1957 and introduced a wide audience to elegant mathematical puzzles and diversions.^[4] Singmaster later expressed admiration for Gardner's autobiographical reflections, crediting the column's enchanting style for inspiring generations of mathematicians to explore puzzles beyond formal academia.^[18] Following his move to the United Kingdom in 1970 to take up a position at the Polytechnic of the South Bank (later London South Bank University), Singmaster's puzzle interests deepened and became integrated into his teaching and personal scholarship. His academic role provided the flexibility to pursue these pursuits alongside his professional duties in discrete mathematics.^[1] In London, he shifted toward more systematic study, attending monthly book fairs and visiting specialist shops to expand his collection, which evolved from modest beginnings into a comprehensive library of over 12,000 volumes on recreational mathematics by the early 2020s.^[4] Central to this development was Singmaster's exposure to historical puzzle texts, which broadened his appreciation for the evolution of recreational mathematics. Works such as Luca Pacioli's De Viribus Quantitatis (c. 1498) and Filippo Calandri's Trattato di Arithmetica (1491) facsimile editions captivated him, revealing the ancient roots of puzzles in mathematical education and entertainment.^[1] This fascination contributed to the adoption of the term "metagrobology" in the 1980s, a variant of the 17th-century word "metagrobolize" meaning to puzzle deeply, which Singmaster helped popularize through his decades of research and publication—a field in which he became a leading figure.^[4]^[19]

Analysis of mechanical puzzles

Singmaster employed systematic techniques to analyze mechanical puzzles, often beginning with physical disassembly to understand their internal mechanisms and construction. For instance, in examining interlocking puzzles such as burrs, he documented step-by-step takedown processes, identifying key notches and joints that allow separation while noting variations in wood types and regional manufacturing differences.^[20] This hands-on approach was complemented by historical tracing, where he cross-referenced patents, catalogs, and period literature to map puzzle evolutions, such as the progression from simple wooden keys in 18th-century puzzle boxes to more complex sliding mechanisms in the 19th century.^[21] His cataloguing efforts formed a cornerstone of his work, culminating in extensive databases and bibliographies that tracked puzzle origins, variants, and underlying mathematics. In his annotated bibliography Sources in Recreational Mathematics, Singmaster compiled over 450 topics spanning centuries, including detailed entries on mechanical puzzles with annotations on inventors, production dates, and mathematical properties like parity and topology.^[20] For sliding puzzles, he emphasized topological invariants, such as the connectivity of piece movements on a grid, and cataloged variants from two-dimensional boards to three-dimensional extensions, highlighting how these alterations affect solvability.^[20] His chronology of recreational mathematics further organized this data temporally, logging key milestones like the 1790 appearance of the six-piece burr puzzle and its refinements by 1875.^[21] A representative example of his analytical depth is the Fifteen Puzzle, a 4x4 sliding grid with numbered tiles and one empty space, which Singmaster dissected through group theory to reveal parity invariants as a core mathematical principle. He explained that the puzzle's solvability depends on the even permutation of tiles relative to the blank space's row and column parity; an odd permutation renders half of all configurations impossible, a concept he traced back to its invention by Noyes Palmer Chapman in 1874, popularized by Sam Loyd, and the ensuing 1880 craze.^[22]^[21]^[23] This analysis not only clarified why the "15-14" swapped version defies solution but also connected the puzzle to broader permutation theory, influencing modern variants.^[21] Similarly, Singmaster's breakdown of the Tower of Hanoi—a stack of disks moved between pegs under size constraints—integrated recursive mathematics and graph theory to model optimal solutions. He detailed how the minimal moves follow a binary pattern, with the puzzle's state graph forming a Sierpinski triangle, and attributed its 1883 invention to Édouard Lucas while noting early precursors in Indian texts.^[21] In his historical tour, he explored evolutions like multi-peg variants mentioned by Lucas in 1889, linking the puzzle's exponential move count (2^n - 1 for n disks) to computational complexity. Through these methods, Singmaster advanced puzzle history by identifying obscure inventors and charting evolutions, such as crediting early German catalogs like Bestelmeier's 1801-1803 Magazin for disseminating burr designs across Europe.^[20] His work rescued forgotten figures like 19th-century puzzle makers from obscurity, fostering a scholarly appreciation for how mechanical puzzles embody topological and combinatorial principles.^[21]

Rubik's Cube contributions

Invention of Singmaster notation

David Singmaster developed what became known as Singmaster notation in late 1978, shortly after acquiring his first Rubik's Cube in August of that year at the International Congress of Mathematicians in Helsinki.^[1]^[24] By December 1978, while working on methods to solve and analyze the Cube, he devised a system using single letters to represent 90-degree clockwise rotations of each face: U for the upper face, D for the down face, L for the left face, R for the right face, F for the front face, and B for the back face.^[22]^[25] The primary rationale for creating this notation was to enable a concise, universal, and intuitive method for recording and communicating sequences of moves in personal notes, teaching sessions, and early publications, without relying on diagrams, numerical coordinates, or ambiguous abstract symbols that might confuse non-specialists. Singmaster drew on his experience with recreational mathematics to select these face-oriented letters, rejecting more complex alternatives like John Conway's color-based system or coordinate notations (e.g., x, y, z axes) due to their lack of clarity on direction and handedness.^[22] The notation evolved rapidly during its initial refinement in autumn 1978, with the addition of a prime symbol (') to denote counterclockwise 90-degree turns and the numeral 2 to indicate 180-degree (half) turns, allowing for efficient description of any basic move. These conventions were formalized and first published in Singmaster's "Notes on Rubik's Magic Cube" in February 1979, marking the system's debut in print.^[22]^[25] Singmaster notation was rapidly adopted as the de facto standard within the emerging Rubik's Cube community by 1979, facilitating the sharing of algorithms and influencing the design of early solving software, instructional guides, and competition rules. Its simplicity and effectiveness ensured its enduring use in cubing worldwide.^[2]^[3]

Solution methods and publications

Singmaster developed a layer-by-layer approach to solving the Rubik's Cube, which systematically builds the puzzle from the bottom layer upward, emphasizing intuitive positioning before orientation and permutation adjustments. This method implicitly aligns the center pieces of the first layer by fixing the color scheme, then focuses on placing the first-layer corners correctly relative to the centers, followed by solving the edges of the middle layer to complete the first two layers. The process concludes with the third layer, where edges are positioned, corners are oriented, and finally both edges and corners are permuted into their solved states, often described in a seven-step framework that avoids advanced commutators for accessibility. In his seminal 1979 publication, Notes on Rubik's Magic Cube, Singmaster provided one of the earliest comprehensive guides to solving the Cube, integrating practical instructions with mathematical exposition.^[24] The booklet introduces basic group theory concepts, portraying the Cube's configurations as elements of a finite group generated by face rotations, and explores permutations of edges and corners, including how sequences of moves cycle these pieces while preserving overall parity. It highlights key challenges like even permutations required for solvability, noting that impossible states arise from odd parity in edge or corner arrangements.^[19] Singmaster's work underscored the Cube's immense scale, calculating the order of its symmetry group as approximately 43 quintillion (4.3 × 10¹⁹) positions, which illustrates the puzzle's combinatorial depth and the efficiency of layer-by-layer strategies in navigating this vast space. He addressed parity issues explicitly, explaining how certain configurations—such as a single edge flip or twisted corner—are unreachable due to the even permutation constraints inherent in quarter-turn moves. Through early writings like Notes on Rubik's Magic Cube and subsequent promotions, Singmaster significantly influenced the burgeoning cubing community by disseminating solution techniques via newsletters such as the Cubic Circular (1981–1985), which shared analyses, news, and puzzle variants to connect enthusiasts worldwide.^[24] Singmaster notation served as a concise tool for documenting these methods in his publications, enabling clear communication of algorithms.^[24]

Publications

Books and monographs

David Singmaster authored several influential books and monographs that bridged academic mathematics with recreational puzzles, emphasizing historical context, problem-solving, and analytical depth. His works often drew from his extensive research into the origins of mathematical recreations, reflecting his career-long interest in making complex ideas accessible through engaging formats. These publications, primarily from the later stages of his career, compiled decades of original contributions and scholarly insights. One of his notable monographs is Problems for Metagrobologists: A Collection of Puzzles with Real Mathematical, Logical or Scientific Content (2016), published by World Scientific. This volume assembles over 200 original problems that Singmaster composed starting in 1987, many of which first appeared in his puzzle columns for periodicals like the Los Angeles Times and Micromath. The puzzles cover diverse topics, including geometry, combinatorics, and logic, designed to challenge readers with genuine mathematical substance while encouraging creative solutions.^[26] Singmaster's two-volume set, Adventures in Recreational Mathematics (Volume I, 2020; Volume II, 2021), also published by World Scientific, offers a deep exploration of recreational mathematics through historical essays and analytical discussions. Drawing on his vast collection of antique puzzles and texts, the books trace the evolution of ancient and medieval recreational problems, such as tangrams, magic squares, and dissection puzzles, back to their earliest documented sources in manuscripts from the 10th to 18th centuries. Volume I focuses on geometric and combinatorial origins, while Volume II delves into numerical and algebraic recreations, providing annotated bibliographies and reproductions of rare illustrations to contextualize their development. Singmaster's approach highlights the interplay between play and rigorous mathematics, making the set a key resource for historians and enthusiasts alike.^[27] Singmaster's early work on the Rubik's Cube includes Notes on Rubik's "Magic Cube" (1979), a self-published guide that provided the first English-language instructions for solving the puzzle and introduced his standard notation system. An expanded edition was published by Enslow in 1981.^[1] In addition to his solo-authored works, Singmaster contributed a significant chapter to the edited volume A Lifetime of Puzzles: Honoring Martin Gardner (2008), published by A K Peters. Titled "De Viribus Quantitatis by Luca Pacioli: The First Recreational Mathematics Book," the chapter examines the 1509 incunable by Luca Pacioli, arguing it as the inaugural printed work dedicated to recreational mathematics, with detailed analysis of its polyhedral models, magic squares, and cryptographic elements. This contribution underscores Singmaster's expertise in the historiography of mathematical recreations.^[28] Among his earlier monographs related to recreational mathematics, Singmaster co-authored Handbook of Cubic Math (1982) with Alexander H. Frey Jr., published by Enslow Publishers, which systematically analyzes the Rubik's Cube's group theory and solution algorithms, serving as a foundational text for understanding the puzzle's mathematical structure.^[29] Singmaster also compiled the Mathematical Gazetteer of the British Isles (2012), a comprehensive catalog of over 1,000 mathematical sites, buildings, and artifacts across the UK and Ireland, originally serialized in the British Society for the History of Mathematics Newsletter from 1995 to 2003.^[30]

Newsletters and articles

Singmaster produced the Cubic Circular, a quarterly newsletter dedicated to the Rubik's Cube and related mechanical puzzles, from autumn 1981 to summer 1985.^[31] It comprised five magazines, three of which were double issues, totaling eight issues that covered cube variants, solving techniques, competition updates, new puzzle products, and mathematical explorations such as n-dimensional cubes and polyhedral generalizations.^[31] The newsletter served as a key resource for the burgeoning cubing community, distributing technical notes and fostering enthusiast correspondence.^[31] In popular science outlets, Singmaster contributed accessible articles on recreational mathematics and puzzles. His piece "Moral and Mathematical Lessons from a Rubik Cube," published in New Scientist on 23/30 December 1982 (pp. 786–791), reflected on the cube's educational value, ethical implications in problem-solving, and broader mathematical insights derived from its mechanics.^[24] Singmaster also authored or co-authored articles in academic journals focused on puzzle analysis. A notable example is "The Jealous Husbands and the Missionaries and Cannibals," co-written with Ian Pressman and appearing in The Mathematical Gazette (Vol. 73, No. 464, June 1989, pp. 73–81), which examined graph-theoretic models and solution paths for these classic river-crossing puzzles, including variations with multiple couples or cannibals.^[32] He contributed problems and shorter pieces to journals such as the American Mathematical Monthly, often posing recreational challenges that highlighted combinatorial or geometric principles.^[33] Notable among his translations is the 1992 article "Problems to Sharpen the Young," co-authored with John Hadley and published in The Mathematical Gazette (Vol. 76, No. 475, March 1992, pp. 102–126), providing the first complete English translation and annotation of the medieval Latin puzzle collection Propositiones ad acuendos juvenes attributed to Alcuin of York.^[34] Additionally, Singmaster circulated unpublished notes and preliminary bibliographies on puzzle histories through personal networks and conferences. His Sources in Recreational Mathematics, an annotated bibliography begun in 1982 with eight preliminary editions by 2004 (spanning 818 pages across ~457 topics), cataloged historical texts on games, puzzles, and recreations from ancient to modern eras, emphasizing chronological development and avoiding standard number theory.^[20] This work, distributed informally via CDs and direct contact, provided a foundational reference for researchers in recreational mathematics.^[20]

Public appearances

Television and media

Singmaster made a notable television appearance on the BBC's science fiction-themed puzzle game show The Adventure Game in 1981. In the episode aired on November 16, he joined presenter Sue Cook and musician Philip Sheppard as celebrity contestants on the fictional planet Arg, where they solved a series of interactive puzzles and challenges to rescue host Lesley Judd from the alien character Ungilon. Leveraging his expertise in recreational mathematics, Singmaster demonstrated puzzle-solving techniques amid the show's adventurous format, which blended live-action role-playing with physical and logical tasks.^[35]^[22] From June 1998 to November 1999, Singmaster served as a frequent panelist on the BBC Radio 4 program Puzzle Panel, a light-hearted half-hour show hosted by Chris Maslanka. Alongside other puzzle enthusiasts, he tackled brain teasers, contributed original puzzles, and discussed recreational mathematics topics, often drawing on listener-submitted problems to highlight clever logical solutions. This role allowed him to promote public interest in puzzles through broadcast media.^[36] Singmaster also participated in various radio interviews focused on puzzle topics, sharing insights into the history and mechanics of recreational mathematics and mechanical puzzles like the Rubik's Cube. His contributions extended to providing puzzle content for BBC radio broadcasts, further disseminating his knowledge to wider audiences.^[19]

Conferences and interviews

David Singmaster was a dedicated participant in the Gathering for Gardner (G4G) conferences, attending every biennial event since the inaugural gathering in 1993 and delivering presentations at many of them.^[4] These conferences, held in honor of mathematician Martin Gardner, provided a platform for Singmaster to engage with fellow puzzle enthusiasts and scholars on topics in recreational mathematics.^[4] His involvement included notable interactions, such as meeting Rubik's Cube inventor Ernő Rubik at a G4G event in Atlanta.^[4] At G4G13 in April 2018, Singmaster featured prominently through interviews that highlighted his contributions to puzzle history.^[37] In one extended interview, he discussed the early development of Rubik's Cube notation, his extensive personal collections of mechanical puzzles, and broader insights into recreational mathematics.^[38] A separate interview focused specifically on the Rubik's Cube, recounting its initial reception and solving methods.^[39] These sessions, recorded and shared via YouTube, underscored his role as a key figure in the puzzle community.^[40] Singmaster also contributed significantly to the British Society for the History of Mathematics (BSHM), where he organized events and delivered at least twelve presentations on recreational mathematics topics.^[3] He coordinated the 1996 "History of Recreational Mathematics" meeting at South Bank University and spoke on subjects ranging from "The Magic Cube" in 1979 to "The 17 camels and the other division problems" at the 2019 Birkbeck meeting.^[3] His talks emphasized historical contexts of puzzles and their mathematical underpinnings.^[3] Throughout his conference appearances, Singmaster exerted a mentoring influence on cubers and puzzle enthusiasts by generously sharing his knowledge, resources, and anecdotes, fostering curiosity and inclusivity among attendees.^[37] At events like G4G13, he engaged directly with participants, offering guidance that left a lasting impact on emerging scholars in the field.^[37]

Personal life and death

Family and collections

David Singmaster was married twice; his second marriage was to Deborah in 1972, whom he met during the underwater archaeological expedition off Sicily.^[1]^[41]^[4] Deborah accompanied him on travels and hosted visitors interested in his collections at their London home. Singmaster and Deborah adopted a daughter, Jessica, in 1976. He generally maintained privacy regarding family matters.^[1] Singmaster amassed a substantial library of approximately 10,000 to 12,000 volumes, focused on recreational mathematics, the history of puzzles, gazetteers of the British Isles, and related topics such as mathematical history, cartoons, humor, and language.^[3]^[4] He began building this collection significantly after relocating to London in 1970, seeking a permanent institutional home for it in later years, such as with the Mathematical Association.^[4] His passion for puzzles extended to a personal collection exceeding 3,000 mechanical puzzles by 2002, including more than 400 variants of the Rubik's Cube, which he acquired and cataloged over half a century starting in the 1960s.^[24]^[4] These items, reflecting his deep interest in metagrobology—the study of puzzles—were housed in a dedicated extension to his home, transforming part of the space into a personal studio for examination and display.^[3]^[41] Singmaster's hobbies included extensive travels to acquire puzzles and conduct historical research, such as a year in Tuscany, Italy, in 1971 with his wife to study mathematical manuscripts, and participation in biennial Gatherings for Gardner conferences in Atlanta starting in 1993 for networking and puzzle exchanges.^[4]^[3] He also joined underwater archaeological expeditions, including one in Sicily in 1971 that uncovered a Punic shipwreck, blending his interests in history and discovery.^[3]

Death and legacy

David Singmaster died on 13 February 2023 in London at the age of 84, following a long illness, though the specific cause was not publicly detailed.^[1]^[37]^[4] His passing prompted widespread tributes within the mathematics and puzzle communities. An obituary by Colin Wright appeared in Chalkdust magazine in May 2023, highlighting Singmaster's passion for puzzles and his generous spirit.^[2] The Gathering for Gardner organization collected personal remembrances from colleagues and friends, emphasizing his role in advancing recreational mathematics.^[37] Cubing enthusiasts shared memories on dedicated forums, such as the site honoring his contributions to Rubik's Cube history.^[42] Singmaster's legacy endures through his standardization of Rubik's Cube notation, which he introduced in 1979 and remains the community standard for describing moves in competitions and tutorials as of 2025.^[43]^[25] His vast personal archives, including extensive puzzle collections and historical documents, have been preserved at London South Bank University's Archives Centre, ensuring access for future researchers.^[44] As a pioneering metagrobologist—the study of mathematical recreations—he inspired generations of puzzle enthusiasts and scholars to explore the intersections of mathematics and play.^[37] Posthumously, Singmaster's methods continue to influence education and competitive cubing; for instance, the World Cube Association employs his notation in official events, and a 2024 workshop on recreational mathematics history was held in his memory by the British Society for the History of Mathematics.^[45] His collections form a cornerstone of this enduring impact, cataloging thousands of puzzles that inform ongoing studies in the field.^[44]

References

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David Singmaster - Biography - MacTutor - University of St Andrews
David Singmaster was an American mathematician who worked for most of his career in London, England. He was a leading expert on puzzles and recreational ...
[2]
Significant figures: David Singmaster (1938–2023) - Chalkdust
May 22, 2023 · David is known for Singmaster's conjecture. In the pattern we know as Pascal's triangle, every number greater than one will appear somewhere.
[3]
Full article: David Singmaster (December 1938 to 13 February 2023)
Jul 24, 2023 · David Singmaster's involvement in history of mathematics, by Tony Mann. David organized and contributed to many BSHM events over the years.Missing: biography | Show results with:biography
[4]
David Singmaster - Sheila Markham
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 ...
[5]
[PDF] Derrick Henry Lehmer
Note that David Singmaster was also a PhD student of R. S. Lehman while DHL was on leave. 1. Donald Marvin Adelman (1947), Some arithmetic properties of ...Missing: supervisor | Show results with:supervisor
[6]
Notes on Binomial Coefficients Iii—Any Integer Divides Almost All ...
Notes on Binomial Coefficients Iii—Any Integer Divides Almost All Binomial Coefficients†. David Singmaster,. David Singmaster. Polytechnic of the South Bank ...
[7]
On polynomial functions (mod m) - ScienceDirect.com
In this paper, we obtain a canonical representation for those polynomials (with integer coefficients) which vanish (mod m), a canonical representation for each ...
[8]
[PDF] divisibility of binomial and multinomial coefficients
DAVID SINGMASTER. Polytechnic of the South Bank, London and. Istituto ... 49. 50. 51. S. Zuckerman. An Introduction to the Theory of Numbers. New York: 1960. John.Missing: publications | Show results with:publications
[9]
https://www.fq.math.ca/Books/Collection/singmaster.pdf
[10]
Singmaster's conjecture - MathOverflow
Jun 19, 2010 · One number is known to appear 8 times. No one knows whether there are any others nor whether any number appears more than 8 times. Singmaster ...Not especially famous, long-open problems which anyone can ...Examples of seemingly elementary problems that are hard to solve?More results from mathoverflow.net
[11]
Singmaster's conjecture in the interior of Pascal's triangle - Terry Tao
Jun 7, 2021 · This paper leverages the theory of exponential sums over primes to make progress on a well known conjecture of Singmaster which asserts that any natural number ...Missing: multiplicity | Show results with:multiplicity
[12]
How often Does an Integer Occur as a Binomial Coefficient?
Apr 11, 2018 · ... How often Does an Integer Occur as a Binomial Coefficient? The American Mathematical Monthly ... David SingmasterPolytechnic of the South Bank, ...
[13]
Open Season – Singmaster's Conjecture | The Aperiodical
Jan 29, 2013 · Singmaster proved in 1971 that the number of occurrences of a is N ( a ) = O ( log ⁡ a ) ; that is, as you start to consider larger and larger ...
[14]
Singmaster's conjecture in the interior of Pascal's triangle - arXiv
Jun 7, 2021 · Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle.Missing: multiplicity | Show results with:multiplicity
[15]
Frequency of numbers in Pascal's triangle - IMA
Dec 13, 2017 · The smallest number that appears 8 times is 3003. No examples have been found of numbers that appear exactly 5 times; however, it is not known ...
[16]
Singmaster's Conjecture In The Interior Of Pascal's Triangle
Apr 5, 2022 · Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that ...
[17]
Mathematics: Master puzzler - Nature
Sep 18, 2013 · David Singmaster delights in the autobiography of Martin Gardner, whose Scientific American maths column enchanted tens of thousands.Missing: biography PhD supervisor
[18]
David Singmaster's books - MacTutor
David Singmaster's books. We give information below about six books by ... My interest in puzzles has always been in understanding, not just the puzzle ...
[19]
Sources in recreational mathematics - The Puzzle Museum
Mar 19, 2004 · SOURCES IN RECREATIONAL MATHEMATICS. AN ANNOTATED BIBLIOGRAPHY. EIGHTH PRELIMINARY EDITION. DAVID SINGMASTER. Contact via the Puzzle Museum.Missing: influence | Show results with:influence
[20]
CHRONOLOGY OF RECREATIONAL MATHEMATICS by David ...
La fonte principale di questa cronologia è la "Chronology of recreational mathematics" di David Singmaster, nella versione del 1998. Un'ultima annotazione ...
[21]
TwistyPuzzles.com > Articles > Interview With David Singmaster
David Singmaster is a Professor of Mathematics at South Bank University, London. He has been a prominent figure in the cubing world since the early days.
[22]
David Singmaster - Speedsolving.com Wiki
May 24, 2023 · David Breyer Singmaster (1938–2023) was an American mathematics professor at London South Bank University in the United Kingdom.
[23]
Notation | Cubing History
The community standard outer turn notation was created by David Singmaster in 1979. In Notes on Rubik's 'Magic Cube', Singmaster uses U, D, L, R, F, and B for ...
[24]
David Singmaster - Twisty Puzzles Wiki - Fandom
Singmaster's association with the Rubik's Cube began in August 1978, when he first encountered the puzzle at the International Congress of Mathematicians in ...
[25]
http://www.cubinghistory.com/3x3/3x3Notation
[26]
https://www.worldscientific.com/worldscibooks/10.1142/9531
[27]
A Lifetime of Puzzles: Honoring Martin Gardner
May 12, 2009 · The articles include puzzles, personal recollections of the authors' friendships with Gardner, and historical pieces. The personalized articles ...Missing: early | Show results with:early
[28]
Cubic Circular
The Cubic Circular magazine was written and produced by David Singmaster in the early Eighties. The first issue was a small 16 page pink booklet.
[29]
The jealous husbands and The missionaries and cannibals
Aug 1, 2016 · The jealous husbands and The missionaries and cannibals - Volume 73 Issue 464. ... David Singmaster. Show author details. Ian Pressman ...
[30]
The American Mathematical Monthly: Vol 78, No 4
The American Mathematical Monthly, Volume 78, Issue 4 (1971). Browse ... How often Does an Integer Occur as a Binomial Coefficient? xml · David Singmaster.
[31]
The Adventure Game, Series 2, Episode 3 - BBC
Trying to rescue Lesley Judd from Arg are David Singmaster, Sue Cook and Philip Sheppard ... Next. Episode 5—Series 2 · See all episodes from The Adventure Game ...
[32]
BBC Radio 4 Extra - Puzzle Panel, Episode 10
BBC Radio 4 Extra. Puzzle Panel. More. Home · Episodes. Main content. Sorry, this episode is not currently available. Episode 10. Puzzle Panel. Episode 10 of 10.
[33]
Remembering David Singmaster - Gathering 4 Gardner
Feb 22, 2023 · It is with great sadness that G4G notes the death of David Singmaster (December 1938 – February 13, 2023). We are collecting here some recollections.Missing: biography | Show results with:biography
[34]
David Singmaster - An Extended Interview - G4G13 Apr 2018
Jun 23, 2018 · David Singmaster - An Extended Interview - G4G13 Apr 2018. 695 views · 7 years ago ...more. G4G Celebration. 4.71K. Subscribe. 16. Share.
[35]
Interview Videos - Gathering 4 Gardner
Dec 14, 2022 · Ken Brecher – Pseudosphere Rolling Uphill – March 23, 2014 · Rubik's Cube – David Singmaster · Sarah Garvin Rodgers - Interview - G4G13 Apr 2018 · Robert Crease - ...Missing: conferences | Show results with:conferences
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David Singmaster - Interview - G4G13 Apr 2018 - YouTube
Jun 19, 2018 · David Singmaster - Interview - G4G13 Apr 2018. 257 views · 7 years ago ...more. G4G Celebration. 4.71K. Subscribe.
[37]
David Singmaster RIP - Rob Eastaway
Feb 15, 2023 · I was running a workshop on creative thinking and puzzles for some engineering students at South Bank University. “You should meet my colleague ...Missing: States Antioch Michigan
[38]
Remembering David Singmaster
Remembering David Singmaster (1938-2023). Father of cubing theory, notation, and layer-by-layer. We lost the legendary David Singmaster in February 2023.Missing: biography | Show results with:biography
[39]
Notation | WikiCube | Fandom
No prime or 2 means 72° clockwise, a 2 means 144°, and a prime means counterclockwise. Each turn has 4 forms (U, U', U2, U2'). Any notation for the Megaminx ...Missing: primes | Show results with:primes
[40]
About The Archives
We also hold archives for the following: David Singmaster (former professor emeritus); Greater London War Risk Study (GLAWARS); Local Endowed Charities; South ...
[41]
Events Archive | The British Society for the History of Mathematics
Recreational Mathematics and its History: A workshop in memory of David Singmaster. Sat 21 September 2024. De Morgan House, London and Online. BSHM-LMS-Glasgow ...