David Singmaster

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David Singmaster
David Singmaster 2006.jpg
March 2006
Born 1939 (age 77–78)
Ferguson, Missouri[1]
Occupation Retired professor of mathematics
Employer London South Bank University
Known for Mathematics of puzzles, especially the Rubik's cube

David Breyer Singmaster (born 1939, USA[1]) is a retired professor of mathematics at London South Bank University, England, UK. A self-described metagrobologist, he has a huge personal collection of mechanical puzzles and books of brain teasers. He is 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 (i.e., letters which denote the faces on the Rubik's Cube) which has become standard. He is both a puzzle historian and a composer of puzzles, and many of his puzzles have been published in newspapers and magazines. In combinatorial number theory, Singmaster's conjecture states that there is a upper bound on the number of times a number other than 1 can appear in Pascal's triangle.


David Singmaster was a student at the California Institute of Technology in the late 1950s.[2] He intended to be a chemist, then changed to physics.[3] He only became really interested in mathematics in his final year when he took some courses in algebra and number theory.[3] 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.[3]

Singmaster was a lecturer in the department of Mathematical Sciences at the Polytechnic of the South Bank in London from the early 1970s.[4] Around 1972 he attended the Istituto di Matematica in Pisa for a year having won a research scholarship.[3] He was promoted to a Readership (a Research Professorship) at the South Bank polytechnic in September 1984.[5] The polytechnic became London South Bank University in 1992, and Singmaster was the professor of mathematics at the "School of Computing, Information Systems and Mathematics". He also became an honorary research fellow at University College London.[6]

Rubik's Cubes[edit]

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 in Helsinki.[3] Some other mathematicians at the conference, including John Conway and Roger Penrose, had already acquired one.[3] Singmaster quickly acquired a Cube (in exchange for a copy of an M. C. Escher book) and was able to solve it by early September 1978.[3] He has said that it took him "two weeks, on and off" to find a general solution for the Cube.[7] He devised his notation for recording moves (now known as the Singmaster notation) in December 1978.[4] In June 1979 he wrote one of the first articles about the Cube in The Observer newspaper.[8]

The power of conjugation ... was the last point I understood; I remember lying awake thinking about it, seeing that I could move any four edges into the working locations and realising that this completed the general method for restoring the cube to its original state.

–David Singmaster, Moral and Mathematical Lessons from a Rubik Cube, New Scientist, 1982

In October 1979 he self-published his Notes on the "Magic Cube".[9] The booklet contained his mathematical analysis of the Cube, allowing a solution to be constructed using basic group theory.[10] In August 1980 he published an expanded 5th edition of the book retitled as Notes on Rubik's "Magic Cube".[9] It included the results of his correspondence with other "cubologists", and included details on monotwists, U-flips, Cayley graphs, and wreath products.[10] The book contained his own "step by step solution" for the Cube,[11] and it is accepted that he was a pioneer of the general Layer by Layer approach for solving the Cube.[12] 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".[13] In 1981, at the height of the Rubik's Cube craze, the book was republished by Penguin Books, with a US edition by Enslow Publishers.[9] There were also Dutch and Spanish translations.[9] He estimates that he sold around 50 to 60 000 copies of his book.[3] Much of the mathematical content of the book was later reworked into his 1982 work Handbook of Cubik Math, co-written with Alexander H. Frey.

Singmaster has been described as "one of the most enthusiastic and prolific promoters of the Cube".[14] In September 1981 he was said to be devoting "almost 100%" of his time to promoting, reporting, marketing and analysing the Cube.[15] He soon began publishing a quarterly newsletter called the Cubic Circular which was published between 1981 and 1985.[3][15]


Singmaster has a huge personal collection of at least 3000 mechanical puzzles, of which about 400 are Rubik's Cubes and variants.[3] From around 1980 to 1982 he ran his own puzzle company, David Singmaster Ltd, which stocked "over 100 puzzles and books".[16] However the venture lost him "a fair amount of money" and led to "prolonged tax negotiations".[5]

Singmaster is both a puzzle historian and a composer of puzzles, and he describes himself as a "metagrobologist". Many of his puzzles have appeared in publications such as BBC Focus, Games & Puzzles, the Los Angeles Times, and the Weekend Telegraph.[17] From around 2006 Singmaster was a director at the Conjuring Arts Research Center, retiring from the position (becoming Director Emeritus) in 2013.[18] 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 (English: On The Powers Of Numbers) was penned by Luca Pacioli, a Franciscan monk who lived around 1500.[19]

Singmaster's conjecture[edit]

In combinatorial number theory, Singmaster's conjecture states that there is a finite upper bound on the number of times a number other than 1 can appear in Pascal's triangle. Paul Erdős suspected that the conjecture is true, but thought it would probably be very difficult to prove. The empirical evidence is consistent with the proposition that the smallest upper bound is 8.

Media appearances[edit]

In November 1981, he appeared on the scifi-themed BBC puzzle show The Adventure Game.[3] From 1998 to 1999 he was a frequent panelist on the BBC Radio 4 show Puzzle Panel.[3]

Personal life[edit]

Singmaster has been married twice, the second time to Deborah in 1972. They have one daughter, Jessica, adopted in 1976.[3]





See also[edit]


  1. ^ a b "David Singmaster in the 1940 Census". ancestry.com. Retrieved 13 January 2016. 
  2. ^ "Candidates' statements - treasurer" (PDF). The California Tech. 20 February 1958. p. 9. 
  3. ^ a b c d e f g h i j k l m "Interview with David Singmaster". Twisty Puzzles. Retrieved 4 January 2016. 
  4. ^ a b Singmaster, David (23 December 1982). "Moral and Mathematical Lesson from a Rubik Cube". New Scientist. p. 787. 
  5. ^ a b David Singmaster (1985). "Cubic Circular Issues 7 & 8". 
  6. ^ "A lecture to get your head around". University College London. 10 January 2007. 
  7. ^ Jensen, Gregory (24 August 1981). "Now meet Rubik's snake --'Bigger than Rubik's cube!'". United Press International. 
  8. ^ David Singmaster (17 June 1979). "Six-sided magic". The Observer. 
  9. ^ a b c d "Publications of David Singmaster". eldar.org. 4 August 1996. 
  10. ^ a b "Review - Restore your cube". New Scientist. 24 September 1981. p. 802. 
  11. ^ David Singmaster (1980-08-06). "A Step by Step Solution of Rubik's "Magic Cube"". Jeffrey W Baumann & LinkedResources. Archived from the original on 2006-03-04. 
  12. ^ Ryan Heise. "Beginner's Rubik's Cube Solution". Archived from the original on 2015-09-26. The general layer-by-layer approach described above is credited to mathematician David Singmaster and was first published in his 1980 book "Notes on Rubik's Magic Cube" 
  13. ^ David Singmaster (8 October 1998). "Davenport's pattern". cube20.org. 
  14. ^ Lees-Maffei, Grace (2015). Iconic Designs: 50 Stories about 50 Things. Bloomsbury. p. 140. ISBN 0857853538. 
  15. ^ a b Herman, Ros (10 September 1981). "Cubic mastery". New Scientist. 
  16. ^ "For Sale". New Scientist. 6 May 1982. p. 395. 
  17. ^ "Problems For Metagrobologists". Telegraph bookshop. Retrieved 4 January 2017. 
  18. ^ Board of Directors, Conjuring Arts. Retrieved 4 January 2017
  19. ^ "And that's renaissance magic ...". The Guardian. 10 April 2007. 

External links[edit]