John Horton Conway

English mathematician (1937–2020)

"John Conway" redirects here. For other uses, see John Conway (disambiguation).

John Horton Conway

FRS

Conway in June 2005

Born| (1937-12-26)26 December 1937

Liverpool, England

Died| 11 April 2020(2020-04-11) (aged 82)

New Brunswick, New Jersey, U.S.

Education| Gonville and Caius College, Cambridge (BA, MA, PhD)
Known for|

* ATLAS of Finite Groups
* Conway chained arrow notation
* Conway criterion
* Conway groups
* Conway notation (knot theory)
* Conway polyhedron notation
* Conway's Game of Life
* Doomsday algorithm
* Free will theorem
* Icosians
* Look-and-say sequence
* Mathieu groupoid
* Monstrous moonshine
* Pinwheel tiling
* Surreal numbers

Awards|

* Berwick Prize (1971)
* Fellow of the Royal Society (1981)
* Pólya Prize (1987)
* Nemmers Prize in Mathematics (1998)
* Leroy P. Steele Prize (2000)

Scientific career
Fields| Mathematics
Institutions| University of Cambridge
Princeton University
Thesis| Homogeneous ordered sets (1964)
Doctoral advisor| Harold Davenport[1]
Doctoral students|

* Richard Borcherds[1]
* Adrian Mathias[1]
* Simon Norton[1]
* Robert Wilson[1]

Website| Archived version @ web.archive.org

John Horton Conway FRS (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life.

Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career.[2][3][4][5][6][7] On 11 April 2020, at age 82, he died of complications from COVID-19.[8]

## Early life and education[edit]

Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce.[7][9] He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician.[10][11] After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge.[9] A "terribly introverted adolescent" in school, he took his admission to Cambridge as an opportunity to transform himself into an extrovert, a change which would later earn him the nickname of "the world's most charismatic mathematician".[12][13]

Conway was awarded a BA in 1959 and, supervised by Harold Davenport, began to undertake research in number theory. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals.[11] It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room.[7]

In 1964, Conway was awarded his doctorate and was appointed as College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge.[14]

After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University.[14] There, he won the school's Pi Day pie-eating contest.[15]

## Conway and Martin Gardner[edit]

Conway's career was intertwined with that of Martin Gardner. When Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity.[16][17] Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work.[18] For instance, he discussed Conway's game of Sprouts (July 1967), Hackenbush (January 1972), and his angel and devil problem (February 1974). In the September 1976 column, he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers.[19]

Conway was a prominent member of Martin Gardner's Mathematical Grapevine. He regularly visited Gardner and often wrote him long letters summarizing his recreational research. In a 1976 visit, Gardner kept him for a week, pumping him for information on the Penrose tilings which had just been announced. Conway had discovered many (if not most) of the major properties of the tilings.[20] Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column.[21] The cover of that issue of Scientific American features the Penrose tiles and is based on a sketch by Conway.[17]

## Personal life and death[edit]

Conway was married three times. With his first two wives he had two sons and four daughters. He married Diana in 2001 and had another son with her.[22] He had three grandchildren and two great-grandchildren.[7]

On 8 April 2020, Conway developed symptoms of COVID-19.[23] On 11 April, he died in New Brunswick, New Jersey, at the age of 82.[23][24][25][26][27]

## Major areas of research[edit]

### Recreational mathematics[edit]

Main article: Conway's Game of Life

A single Gosper's Glider Gun creating "gliders" in Conway's Game of Life

Conway invented the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, long before personal computers existed. Since Conway's game was popularized by Martin Gardner in Scientific American in 1970,[28] it has spawned hundreds of computer programs, web sites, and articles.[29] It is a staple of recreational mathematics. There is an extensive wiki devoted to curating and cataloging the various aspects of the game.[30] From the earliest days, it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. Conway came to dislike how discussions of him heavily focused on his Game of Life, feeling that it overshadowed deeper and more important things he had done, although he remained proud of his work on it.[31] The game helped to launch a new branch of mathematics, the field of cellular automata.[32] The Game of Life is known to be Turing complete.[33][34]

### Combinatorial game theory[edit]

Conway contributed to combinatorial game theory (CGT), a theory of partisan games. He developed the theory with Elwyn Berlekamp and Richard Guy, and also co-authored the book Winning Ways for your Mathematical Plays with them. He also wrote On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT.

He was also one of the inventors of the game sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conway's soldiers. He came up with the angel problem, which was solved in 2006.

He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novelette by Donald Knuth.[35] He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG.

### Geometry[edit]

In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polychoron.[36] Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.

In the theory of tessellations, he devised the Conway criterion which is a fast way to identify many prototiles that tile the plane.[37]

He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice.

### Geometric topology[edit]

In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial.[38] After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.[39] Conway further developed tangle theory and invented a system of notation for tabulating knots, now known as Conway notation, while correcting a number of errors in the 19th-century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings.[40] The Conway knot is named after him.

Conway's conjecture that, in any thrackle, the number of edges is at most equal to the number of vertices, is still open.

### Group theory[edit]

He was the primary author of the ATLAS of Finite Groups giving properties of many finite simple groups. Working with his colleagues Robert Curtis and Simon P. Norton he constructed the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated the Conway groups.[41] This work made him a key player in the successful classification of the finite simple groups.

Based on a 1978 observation by mathematician John McKay, Conway and Norton formulated the complex of conjectures known as monstrous moonshine. This subject, named by Conway, relates the monster group with elliptic modular functions, thus bridging two previously distinct areas of mathematics—finite groups and complex function theory. Monstrous moonshine theory has now been revealed to also have deep connections to string theory.[42]

Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points.

### Number theory[edit]

As a graduate student, he proved one case of a conjecture by Edward Waring, that every integer could be written as the sum of 37 numbers each raised to the fifth power, though Chen Jingrun solved the problem independently before Conway's work could be published.[43] In 1972, Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable. Related to that, he developed the esoteric programming language FRACTRAN. While lecturing on Collatz conjecture, Terence Tao (who was taught by him in graduate school) mentioned Conway's result and said that he was "always very good at making extremely weird connections in mathematics".[44]

### Algebra[edit]

Conway wrote a textbook on Stephen Kleene's theory of state machines and published original work on algebraic structures, focusing particularly on quaternions and octonions.[45] Together with Neil Sloane, he invented the icosians.[46]

### Analysis[edit]

He invented a base 13 function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line, so it has a Darboux property but is not continuous.

### Algorithmics[edit]

For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for anyone with basic arithmetic ability to do the calculations mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practised his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on. One of his early books was on finite-state machines.

### Theoretical physics[edit]

In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a version of the "no hidden variables" principle of quantum mechanics. It states that given certain conditions, if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins to make the measurements consistent with physical law. Conway said that "if experimenters have free will, then so do elementary particles."[47]

## Awards and honours[edit]

Conway received the Berwick Prize (1971),[48] was elected a Fellow of the Royal Society (1981),[49][50] became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS) (1987),[48] won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society. In 2001 he was awarded an honorary degree from the University of Liverpool,[51] and in 2014 one from Alexandru Ioan Cuza University.[52]

His FRS nomination, in 1981, reads:

> A versatile mathematician who combines a deep combinatorial insight with algebraic virtuosity, particularly in the construction and manipulation of "off-beat" algebraic structures which illuminate a wide variety of problems in completely unexpected ways. He has made distinguished contributions to the theory of finite groups, to the theory of knots, to mathematical logic (both set theory and automata theory) and to the theory of games (as also to its practice).[49]

In 2017 Conway was given honorary membership of the British Mathematical Association.[53]

Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, and Conway himself was often a featured speaker at these events, discussing various aspects of recreational mathematics.[54][55]

## Select publications[edit]

Scholia has a profile for John Horton Conway (Q268961).

* 1971 – Regular algebra and finite machines. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series, ISBN 0412106205.
* 1976 – On numbers and games. Academic Press, New York, 1976, Series: L.M.S. monographs, 6, ISBN 0121863506.
* 1979 – On the Distribution of Values of Angles Determined by Coplanar Points (with Paul Erdős, Michael Guy, and H. T. Croft). Journal of the London Mathematical Society, vol. II, series 19, pp. 137–143.
* 1979 – Monstrous Moonshine (with Simon P. Norton).[56] Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308–339.
* 1982 – Winning Ways for your Mathematical Plays (with Richard K. Guy and Elwyn Berlekamp). Academic Press, ISBN 0120911507.
* 1985 – Atlas of finite groups (with Robert Turner Curtis, Simon Phillips Norton, Richard A. Parker, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985, ISBN 0198531990.
* 1988 – Sphere Packings, Lattices, and Groups[57] (with Neil Sloane). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290, ISBN 9780387966175.
* 1995 – Minimal-Energy Clusters of Hard Spheres (with Neil Sloane, R. H. Hardin, and Tom Duff). Discrete & Computational Geometry, vol. 14, no. 3, pp. 237–259.
* 1996 – The Book of Numbers (with Richard K. Guy). Copernicus, New York, 1996, ISBN 0614971667.
* 1997 – The Sensual (quadratic) Form (with Francis Yein Chei Fung). Mathematical Association of America, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26, ISBN 1614440255.
* 2002 – On Quaternions and Octonions (with Derek A. Smith). A. K. Peters, Natick, MA, 2002, ISBN 1568811349.
* 2008 – The Symmetries of Things (with Heidi Burgiel and Chaim Goodman-Strauss). A. K. Peters, Wellesley, MA, 2008, ISBN 1568812205.

## See also[edit]

* List of things named after John Horton Conway

## References[edit]

1. ^ a b c d e John Horton Conway at the Mathematics Genealogy Project
2. ^ Conway, J. H.; Hardin, R. H.; Sloane, N. J. A. (1996). "Packing Lines, Planes, etc.: Packings in Grassmannian Spaces". Experimental Mathematics. 5 (2): 139. arXiv:math/0208004. doi:10.1080/10586458.1996.10504585. S2CID 10895494.
3. ^ Conway, J. H.; Sloane, N. J. A. (1990). "A new upper bound on the minimal distance of self-dual codes". IEEE Transactions on Information Theory. 36 (6): 1319. doi:10.1109/18.59931.
4. ^ Conway, J. H.; Sloane, N. J. A. (1993). "Self-dual codes over the integers modulo 4". Journal of Combinatorial Theory, Series A. 62: 30–45. doi:10.1016/0097-3165(93)90070-O.
5. ^ Conway, J.; Sloane, N. (1982). "Fast quantizing and decoding and algorithms for lattice quantizers and codes" (PDF). IEEE Transactions on Information Theory. 28 (2): 227. CiteSeerX 10.1.1.392.249. doi:10.1109/TIT.1982.1056484.
6. ^ Conway, J. H.; Lagarias, J. C. (1990). "Tiling with polyominoes and combinatorial group theory". Journal of Combinatorial Theory, Series A. 53 (2): 183. doi:10.1016/0097-3165(90)90057-4.
7. ^ a b c d J J O'Connor and E F Robertson (2004). "John Conway – Biography". MacTutor History of Mathematics. Retrieved 24 May 2022.
8. ^ "COVID-19 Kills Renowned Princeton Mathematician, 'Game Of Life' Inventor John Conway In 3 Days". Mercer Daily Voice. 12 April 2020. Retrieved 25 November 2020.
9. ^ a b "CONWAY, Prof. John Horton". Who's Who 2014, A & C Black, an imprint of Bloomsbury Publishing plc, 2014; online edn, Oxford University Press.(subscription required)
10. ^ "John Horton Conway". Dean of the Faculty, Princeton University. Archived from the original on 16 March 2019. Retrieved 3 November 2020.
11. ^ a b Mathematical Frontiers. Infobase Publishing. 2006. p. 38\. ISBN 978-0-7910-9719-9.
12. ^ Roberts, Siobhan (23 July 2015). "John Horton Conway: the world's most charismatic mathematician". The Guardian.
13. ^ Mark Ronan (18 May 2006). Symmetry and the Monster: One of the greatest quests of mathematics. Oxford University Press, UK. pp. 163. ISBN 978-0-19-157938-7.
14. ^ a b Sooyoung Chang (2011). Academic Genealogy of Mathematicians. World Scientific. p. 205\. ISBN 978-981-4282-29-1.
15. ^ "This Is How the Number 3.14 Got the Name 'Pi'". Time. Retrieved 21 September 2022.
16. ^ Mulcahy, Colm (21 October 2014) Martin Gardner, puzzle master extraordinaire, BBC News Magazine: "The Game of Life appeared in Scientific American in 1970, and was by far the most successful of Gardner's columns, in terms of reader response."
17. ^ a b Mulcahy, Colm (21 October 2014). "The Top 10 Martin Gardner Scientific American Articles". Scientific American.
18. ^ The Math Factor Podcast Website John H. Conway reminisces on his long friendship and collaboration with Martin Gardner.
19. ^ Gardner, Martin (1989) Penrose Tiles to Trapdoor Ciphers, W. H. Freeman & Co., ISBN 0-7167-1987-8, Chapter 4. A non-technical overview; reprint of the 1976 Scientific American article.
20. ^ Jackson, Allyn (2005). "Interview with Martin Gardner" (PDF). Notices of the AMS. 52 (6): 602–611.
21. ^ Roberts, Siobhan (28 August 2015). "A Life In Games: The Playful Genius of John Conway". Quanta Magazine.
22. ^ Zandonella, Catherine. "Mathematician John Horton Conway, a 'magical genius' known for inventing the 'Game of Life,' dies at age 82".
23. ^ a b Levine, Cecilia (12 April 2020). "COVID-19 Kills Renowned Princeton Mathematician, 'Game Of Life' Inventor John Conway In 3 Days". Mercer Daily Voice.
24. ^ Zandonella, Catherine (14 April 2020). "Mathematician John Horton Conway, a 'magical genius' known for inventing the 'Game of Life,' dies at age 82". Princeton University. Retrieved 15 April 2020.
25. ^ Van den Brandhof, Alex (12 April 2020). "Mathematician Conway was a playful genius and expert on symmetry". NRC Handelsblad (in Dutch). Retrieved 12 April 2020.
26. ^ Roberts, Siobhan (15 April 2020). "John Horton Conway, a 'Magical Genius' in Math, Dies at 82". New York Times. Retrieved 17 April 2020.
27. ^ Mulcahy, Colm (23 April 2020). "John Horton Conway obituary". The Guardian. ISSN 0261-3077. Retrieved 30 May 2020.
28. ^ Gardner, Martin (October 1970). "Mathematical Games: The fantastic combinations of John Conway's new solitaire game "Life"" (PDF). Scientific American. Vol. 223\. pp. 120–123. JSTOR 24927642.
29. ^ "DMOZ: Conway's Game of Life: Sites". Archived from the original on 17 March 2017. Retrieved 11 January 2017.
30. ^ "LifeWiki". www.conwaylife.com.
31. ^ Does John Conway hate his Game of Life? (video). Youtube
32. ^ MacTutor History: The game made Conway instantly famous, but it also opened up a whole new field of mathematical research, the field of cellular automata.
33. ^ Rendell, Paul (July 2015). Turing Machine Universality of the Game of Life. Emergence, Complexity and Computation. Vol. 18\. Springer. doi:10.1007/978-3-319-19842-2. ISBN 978-3319198415.
34. ^ Case, James (1 April 2014). "Martin Gardner's Mathematical Grapevine". SIAM NEWS. Book reviews of Gardner, Martin, 2013 Undiluted Hocus-Pocus: The Autobiography of Martin Gardner. Princeton University Press and Henle, Michael; Hopkins, Brian (edts.) 2012 Martin Gardner in the Twenty-First Century. MAA Publications.
35. ^ Infinity Plus One, and Other Surreal Numbers by Polly Shulman, Discover Magazine, 1 December 1995
36. ^ Conway, J. H. (1967). "Four-dimensional Archimedean polytopes". Proc. Colloquium on Convexity, Copenhagen. Kobenhavns Univ. Mat. Institut: 38–39.
37. ^ Rhoads, Glenn C. (2005). "Planar tilings by polyominoes, polyhexes, and polyiamonds". Journal of Computational and Applied Mathematics. 174 (2): 329–353. Bibcode:2005JCoAM.174..329R. doi:10.1016/j.cam.2004.05.002.
38. ^ Conway Polynomial Wolfram MathWorld
39. ^ Livingston, Charles (1993) Knot Theory. MAA Textbooks. ISBN 0883850273
40. ^ Perko, Ken (1982). "Primality of certain knots" (PDF). Topology Proceedings. 7: 109–118.
41. ^ Harris, Michael (2015). Review of Genius At Play: The Curious Mind of John Horton Conway. "Mathematics: The mercurial mathematician". Nature. 523 (7561): 406–7. Bibcode:2015Natur.523..406H. doi:10.1038/523406a.
42. ^ Darling, David. Monstrous Moonshine conjecture. Encyclopedia of Science
43. ^ Jorge Nuno Silva (September 2005). "Breakfast with John Horton Conway" (PDF). EMS Newsletter. 57: 32–34.
44. ^ Day 2 - The notorious Collatz conjecture - Terence Tao, retrieved 23 March 2023
45. ^ Conway, John; Smith, Derek A. (2005). "On quaternions and Octonions : their Geometry, Arithmetic, and Symmetry". Bull. Amer. Math. Soc. 42 (2): 229–243. doi:10.1090/S0273-0979-05-01043-8. ISBN 1568811349.
46. ^ Baez, John (2 October 1993). "This Week's Finds in Mathematical Physics (Week 20)".
47. ^ Conway's Proof Of The Free Will Theorem Archived 25 November 2017 at the Wayback Machine by Jasvir Nagra
48. ^ a b "List of LMS prize winners | London Mathematical Society". www.lms.ac.uk.
49. ^ a b "John Conway". The Royal Society. Retrieved 11 April 2020.
50. ^ Curtis, Robert Turner (2022). "John Horton Conway. 26 December 1937—11 April 2020". Biographical Memoirs of Fellows of the Royal Society. 72: 117–138. doi:10.1098/rsbm.2021.0034. S2CID 245355088.
51. ^ Sturla, Anna (14 April 2020). "John H. Conway, a renowned mathematician who created one of the first computer games, dies of coronavirus complications". CNN. Retrieved 16 April 2020.
52. ^ "Doctor Honoris Causa for John Horton Conway". Alexandru Ioan Cuza University. 19 June 2014. Retrieved 7 July 2020.
53. ^ "Honorary Members". The Mathematical Association. Retrieved 11 April 2020.
54. ^ Presentation Videos Archived 9 August 2016 at the Wayback Machine from 2014 Gathering 4 Gardner
55. ^ Bellos, Alex (2008). The science of fun. The Guardian, 30 May 2008
56. ^ Conway, J. H.; Norton, S. P. (1 October 1979). "Monstrous Moonshine". Bulletin of the London Mathematical Society. 11 (3): 308–339. doi:10.1112/blms/11.3.308 - via academic.oup.com.
57. ^ Guy, Richard K. (1989). "Review: Sphere packings, lattices and groups, by J. H. Conway and N. J. A. Sloane" (PDF). Bulletin of the American Mathematical Society. New Series. 21 (1): 142–147. doi:10.1090/s0273-0979-1989-15795-9.

## Sources[edit]

* Alpert, Mark (1999). Not Just Fun and Games Scientific American, April 1999
* Boden, Margaret (2006). Mind As Machine, Oxford University Press, 2006, p. 1271
* du Sautoy, Marcus (2008). Symmetry, HarperCollins, p. 308
* Guy, Richard K (1983). Conway's Prime Producing Machine Mathematics Magazine, Vol. 56, No. 1 (Jan. 1983), pp. 26–33
* Roberts, Siobhan (2015). Genius at play: The curious mind of John Horton Conway. Bloomsbury. ISBN 978-1620405932.
* O'Connor, John J.; Robertson, Edmund F., "John Horton Conway", MacTutor History of Mathematics Archive, University of St Andrews
* John Horton Conway at the Mathematics Genealogy Project
* Princeton University (2009). Bibliography of John H. Conway Mathematics Department
* Seife, Charles (1994). Impressions of Conway The Sciences
* Schleicher, Dierk (2011), Interview with John Conway, Notices of the AMS

## External links[edit]

John Horton Conway at Wikipedia's sister projects

* Media from Commons
* News from Wikinews
* Quotations from Wikiquote

* John Horton Conway's publications indexed by the Scopus bibliographic database. (subscription required)
* Conway, John (20 April 2009). "Proof of the Free Will Theorem" (Video). Archived Lectures.
* John Conway. Videos. Numberphile. playlist on YouTube
* Look-and-Say Numbers. Feat John Conway (2014) on YouTube
* Inventing the Game of Life (2014) on YouTube
* The Princeton Brick (2014) on YouTube Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals and on sums of powers and the Bernoulli numbers
* necrology by Keith Hartnett in Quanta Magazine, April 20, 2020

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