David Singmaster
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David Breyer Singmaster (14 December 1938 – 13 February 2023) was an American-British mathematician who was emeritus professor of mathematics at
London South Bank University London South Bank University (LSBU) is a public university in Elephant and Castle, London. It is based in the London Borough of Southwark, near the South Bank of the River Thames, from which it takes its name. Founded in 1892 as the Borough Po ...
, England. He had a huge personal collection of
mechanical puzzle A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces in which the solution is to manipulate the whole object or parts of it. While puzzles of this type have been in use by humanity as early as the 3rd century BC ...
s and books of
brain teaser A brain teaser is a form of puzzle that requires thought to solve. It often requires thinking in unconventional ways with given constraints in mind; sometimes it also involves lateral thinking. Logic puzzles and riddles are specific types of ...
s. He was most famous for being an early adopter and enthusiastic promoter of the Rubik's Cube. His ''Notes on Rubik's "Magic Cube"'' which he began compiling in 1979 provided the first mathematical analysis of the Cube as well as providing one of the first published solutions. The book contained his cube notation which allowed the recording of Rubik's Cube moves, and which quickly became the standard. Singmaster was both a puzzle historian and a composer of puzzles, and many of his puzzles were published in newspapers and magazines. In combinatorial number theory, Singmaster's conjecture states that there is an upper bound on the number of times a number other than 1 can appear in
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
.


Career

David Singmaster was a student at the
California Institute of Technology The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small group of institutes ...
in the late 1950s. His intention was to become a civil engineer, but he became interested in chemistry and then physics. However he was thrown out of college in his third year for "lack of academic ability". After a year working, he switched to the
University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...
. He only became really interested in mathematics in his final year when he took some courses in
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. In the autumn semester, his number theory teacher Dick Lehmer posed a prize problem which Singmaster won. In his last semester, his algebra teacher posed a question the teacher didn't know the answer to and Singmaster solved it, eventually leading to two papers. He gained his
PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...
from Berkeley, in 1966. He taught at the
American University of Beirut The American University of Beirut (AUB; ) is a private, non-sectarian, and independent university chartered in New York with its main campus in Beirut, Lebanon. AUB is governed by a private, autonomous board of trustees and offers programs le ...
, and then lived for a while in
Cyprus Cyprus (), officially the Republic of Cyprus, is an island country in the eastern Mediterranean Sea. Situated in West Asia, its cultural identity and geopolitical orientation are overwhelmingly Southeast European. Cyprus is the List of isl ...
. Singmaster moved to
London London is the Capital city, capital and List of urban areas in the United Kingdom, largest city of both England and the United Kingdom, with a population of in . London metropolitan area, Its wider metropolitan area is the largest in Wester ...
in 1970. The "Polytechnic of the South Bank" had been created from a merger of institutions in 1970, and Singmaster became a lecturer in the Department of Mathematical Sciences. His academic interests were in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. In August 1971 he joined an archaeological expedition off the coast of
Sicily Sicily (Italian language, Italian and ), officially the Sicilian Region (), is an island in the central Mediterranean Sea, south of the Italian Peninsula in continental Europe and is one of the 20 regions of Italy, regions of Italy. With 4. ...
, acting as photographer. He went off course one day and noticed a timber sticking up out of the sand. This led to the discovery of the Marsala Punic Shipwreck. Around 1972, he attended the Istituto di Matematica in
Pisa Pisa ( ; ) is a city and ''comune'' (municipality) in Tuscany, Central Italy, straddling the Arno just before it empties into the Ligurian Sea. It is the capital city of the Province of Pisa. Although Pisa is known worldwide for the Leaning Tow ...
for a year having won a research scholarship. He was promoted to a Readership (a Research Professorship) at the South Bank Polytechnic in September 1984. The polytechnic college became
London South Bank University London South Bank University (LSBU) is a public university in Elephant and Castle, London. It is based in the London Borough of Southwark, near the South Bank of the River Thames, from which it takes its name. Founded in 1892 as the Borough Po ...
in 1992, and Singmaster was the professor of mathematics at the "School of Computing, Information Systems and Mathematics". He retired in 1996. He became an honorary research fellow at
University College London University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
. He was designated
emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". In some c ...
at
London South Bank University London South Bank University (LSBU) is a public university in Elephant and Castle, London. It is based in the London Borough of Southwark, near the South Bank of the River Thames, from which it takes its name. Founded in 1892 as the Borough Po ...
in 2020.


Rubik's Cubes

Singmaster's association with Rubik's Cubes dates from August 1978, when he saw a Cube (at that time a rarity) at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...
in
Helsinki Helsinki () is the Capital city, capital and most populous List of cities and towns in Finland, city in Finland. It is on the shore of the Gulf of Finland and is the seat of southern Finland's Uusimaa region. About people live in the municipali ...
. Some other mathematicians at the conference, including John Conway and
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
, already had one. Singmaster quickly acquired a Cube (in exchange for a copy of an
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
book) and was able to solve it by early September 1978. He said that it took him "two weeks, on and off" to find a general solution for the Cube. He devised his notation for recording moves (now known as the Singmaster notation) in December 1978. In June 1979 he wrote one of the first articles about the Cube in ''
The Observer ''The Observer'' is a British newspaper published on Sundays. First published in 1791, it is the world's oldest Sunday newspaper. In 1993 it was acquired by Guardian Media Group Limited, and operated as a sister paper to ''The Guardian'' ...
'' newspaper. In October 1979, he self-published his ''Notes on the "Magic Cube"''. The booklet contained his mathematical analysis of Rubik's Cube, allowing a solution to be constructed using basic
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. In August 1980 he published an expanded 5th edition of the book retitled as ''Notes on Rubik's "Magic Cube"''. It included the results of his correspondence with other "cubologists", and included details on monotwists, U-flips,
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...
s, and wreath products. The book contained his own "step by step solution" for the Cube, and it is accepted that he was a pioneer of the general Layer by Layer approach for solving the Cube. The book also contained a catalogue of pretty patterns including his "cube in a cube in a cube" pattern which he had discovered himself "and was very pleased with". In 1981, at the height of the Rubik's Cube craze, the book was republished by
Penguin Books Penguin Books Limited is a Germany, German-owned English publishing, publishing house. It was co-founded in 1935 by Allen Lane with his brothers Richard and John, as a line of the publishers the Bodley Head, only becoming a separate company the ...
, with a US edition by Enslow Publishers. There were also Dutch and Spanish translations. He estimated that he sold around 50 to 60 000 copies of his book. Much of the mathematical content of the book was later reworked by Alexander H. Frey in collaboration with Singmaster to create their ''Handbook of Cubik Math'' published in 1982. Singmaster was described as "one of the most enthusiastic and prolific promoters of the Cube". In September 1981 he was said to be devoting "almost 100%" of his time to promoting, reporting, marketing and analysing the Cube. He soon began publishing a quarterly newsletter called the ''Cubic Circular'' which was published between 1981 and 1985.


Puzzles

Singmaster had one of the world's largest collections of books on recreational mathematics which he had accumulated starting in the late 1970s. In 1996 he reported that the collection contained over 4700 works. He also collected books on cartoons, humour, and language. In 2013 his book collection was reported to be "nearly 10000 items". Many of the books were housed in a library added as an extension to Singmaster's study. He had a huge collection of
mechanical puzzle A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces in which the solution is to manipulate the whole object or parts of it. While puzzles of this type have been in use by humanity as early as the 3rd century BC ...
s, which he started in 2002 containing "perhaps 3000 puzzles, of which about 400 are about Rubik's Cube and its variants". From around 1980 to 1982, he ran his own puzzle company, David Singmaster Ltd, which stocked "over 100 puzzles and books". However the venture lost him "a fair amount of money" and led to prolonged tax negotiations. He referred to this period of his life as "a massive overdose of cubism". Singmaster was both a puzzle historian and a composer of puzzles, and he described himself as a "metagrobologist". Many of his puzzles appeared in publications such as '' BBC Focus'', '' Games & Puzzles'', the ''
Los Angeles Times The ''Los Angeles Times'' is an American Newspaper#Daily, daily newspaper that began publishing in Los Angeles, California, in 1881. Based in the Greater Los Angeles city of El Segundo, California, El Segundo since 2018, it is the List of new ...
'', and the '' Weekend Telegraph''. He published a collection of his puzzles in his 2016 book ''Problems for Metagrobologists''. From around 2006 Singmaster was a director at the New York-based Conjuring Arts Research Center, retiring from the position (becoming Director Emeritus) in 2013. He was instrumental in the re-discovery of one of the world's oldest books on puzzles and magic illusions when he came across a reference to the work in a 19th-century manuscript. The recovered text, ''De viribus quantitatis'' (') was penned by
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
, a Franciscan friar who lived around 1500.


Singmaster's conjecture

In combinatorial number theory, Singmaster's conjecture states that there is a finite
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less ...
on the number of times a number other than 1 can appear in
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
.
Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
suspected that the conjecture is true, but thought it would probably be very difficult to prove. The
empirical evidence Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how the ...
is consistent with the proposition that the smallest upper bound is 8.


Media appearances

In November 1981, Singmaster appeared on the scifi-themed
BBC The British Broadcasting Corporation (BBC) is a British public service broadcaster headquartered at Broadcasting House in London, England. Originally established in 1922 as the British Broadcasting Company, it evolved into its current sta ...
puzzle show '' The Adventure Game''. From 1998 to 1999 he was a frequent panelist on the
BBC Radio 4 BBC Radio 4 is a British national radio station owned and operated by the BBC. The station replaced the BBC Home Service on 30 September 1967 and broadcasts a wide variety of spoken-word programmes from the BBC's headquarters at Broadcasti ...
show '' Puzzle Panel''.


Personal life and death

Singmaster was married twice, the second time to Deborah in 1972. They had one daughter, Jessica, adopted in 1976. Singmaster died on 13 February 2023, at the age of 84.


Publications


Books

* ''Notes on Rubik's "Magic Cube"'', David Singmaster. Enslow Publishers, 1981. * ''Handbook of Cubik Math'', by David Singmaster and Alexander H. Frey. The Lutterworth Press, 1982. . Publisher's description * ''Rubik's Cubic Compendium'', by Ernő Rubik and four others. Edited with an Introduction and Afterword by David Singmaster.
Oxford University Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
, 1987. * ''The Cube: The Ultimate Guide to the World's Bestselling Puzzle'', Jerry Slocum, David Singmaster, Wei-Hwa Huang, Dieter Gebhardt, Geert Hellings, Ernő Rubik. Black Dog & Leventhal, 2009. * ''Problems for Metagrobologists'', David Singmaster, World Scientific Publishing Company, 23 April 2016. * ''Adventures in Recreational Mathematics (in 2 Volumes)''. David Singmaster. World Scientific Publishing Company. (2021)


Reference works

*''Chronology of Recreational Mathematics'' by David Singmaster. 1996.
Available online
at anduin.eldar.org) *''Chronology of Computing'' by David Singmaster. 2000.
Available online
at the University of Applied Sciences, Darmstadt) *''Sources in Recreational Mathematics: An Annotated Bibliography'', David Singmaster. 8th preliminary edition. South Bank University. 2004.

at the Puzzle Museum) *''Mathematical Gazetteer of the British Isles'', by David Singmaster. The British Society for the History of Mathematics. 2012.

at the Internet Archive)


Newsletters

*''Cubic Circular'' magazine published 1981-5 by David Singmaster


Articles


Moral and Mathematical Lessons from a Rubik Cube
by David Singmaster,
New Scientist ''New Scientist'' is a popular science magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organ ...
, 23/30 December 1982 *''The Unreasonable Utility of Recreational Mathematics'' by David Singmaster. First European Congress of Mathematics, Paris, July 1992.
Available online
at anduin.eldar.org) *

'', by David Singmaster and Andrew Southern. Meffert's Puzzles, 15 May 1997.


See also

* ''How to solve the Rubik's Cube'' – Wikibook


References


External links

*
Interview with David Singmaster
at ''Twisty Puzzles''. Originally published c. April 2002

.

A compilation of materials by David Singmaster for teaching and his own interests. Last updated in 1996.
David Singmaster
archive at
London South Bank University London South Bank University (LSBU) is a public university in Elephant and Castle, London. It is based in the London Borough of Southwark, near the South Bank of the River Thames, from which it takes its name. Founded in 1892 as the Borough Po ...
. {{DEFAULTSORT:Singmaster, David 1938 births 2023 deaths Recreational mathematicians Mathematics popularizers Rubik's Cube California Institute of Technology alumni University of California, Berkeley alumni American expatriates in England 20th-century British mathematicians 21st-century British mathematicians Academics of London South Bank University People from St. Louis County, Missouri