John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the Game of Life (the cellular automaton, not the board game).
Conway is currently professor of mathematics at Princeton University. He studied at Cambridge, where he started research under Harold Davenport. He has an Erdős number of one. He received the Berwick Prize (1971), was elected a Fellow of the Royal Society (1981), and was the first recipient of the Pólya Prize (LMS) (1987).
Conway's parents were Agnes Boyce and Cyril Horton Conway. John had two older sisters, Sylvia and Joan. Cyril Conway was a chemistry laboratory assistant. John became interested in mathematics at a very early age and his mother Agnes recalled that he could recite the powers of two when aged four years. John's young years were difficult for he grew up in Britain at a time of wartime shortages. At primary school John was outstanding and he topped almost every class. At the age of eleven his ambition was to become a mathematician.
After leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. He was awarded his BA in 1959 and began to undertake research in number theory supervised by Harold Davenport. 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. It appears that his interest in games began during his years studying at Cambridge, where he became an avid backgammon player spending hours playing the game in the common room. He was awarded his doctorate in 1964 and was appointed as Lecturer in Study at the University of Cambridge.
He left Cambridge in 1986 to take up the appointment to the John von Neumann Chair of Mathematics at Princeton University. He is also a regular visitor at Mathcamp and MathPath, summer math programs for high schoolers and middle schoolers, respectively.
Conway resides in Princeton, New Jersey, United States with his wife and youngest son. He has six other children from his two previous marriages and three grandchildren.
Among amateur mathematicians, he is perhaps most widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy.
He is also one of the inventors of 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 novel by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation.
He is also known for the invention of the Game of Life, one of the early and still celebrated examples of a cellular automaton.
In the mid-1960s with Michael Guy, son of Richard Guy, he established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.
Conway's approach to computing the Alexander polynomial of knot theory involved skein relations, by a variant now called the Alexander-Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed tangle theory and invented a system of notation for tabulating knots.
He worked on the classification of finite simple groups and discovered the Conway groups. He was the primary author of the Atlas of Finite Groups giving properties of many finite simple groups. He with collaborators constructed the first concrete representations of some of the sporadic groups.
With Simon Norton he formulated the complex of conjectures relating the monster group with modular functions, which was christened monstrous moonshine by them.
He has also done work in algebra particularly with quaternions.
For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for anyone with basic arithemetic ability to do the calculations mentally. Conway can usually give the correct answer in under two seconds. To improve his speed, he practices his calendrical calculations on his computer, which is programmed to quiz him with random dates every time he logs on. One of his early books was on finite state machines.
In 2004, Conway and Simon Kochen, another Princeton mathematician, proved the Free will theorem, a startling 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 in order to make the measurements consistent with physical law. Or, in Conway's provocative wording, if experimenters have free will, then so do elementary particles.
He has (co-)written several books including the Atlas of Finite Groups, Regular Algebra and Finite Machines, Sphere Packings, Lattices and Groups, The Sensual (Quadratic) Form, On Numbers and Games, Winning Ways for your Mathematical Plays, The Book of Numbers, and On Quaternions and Octonions. He is currently co-writing The Triangle Book with Steve Sigur, math teacher at Paideia School in Atlanta Georgia, and The Symmetries of Things with Chaim Goodman-Strauss and Heidi Burgiel.
- Charles Seife, "Impressions of Conway", The Sciences
- Mark Alpert, "Not Just Fun and Games", Scientific American April 1999. (official online version; registration-free online version)
- Jasvir Nagra, "Conway's Proof Of The Free Will Theorem" 
- Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
- Video of Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals, and lecturing on sums of powers and Bernoulli numbers.
- Photos of John Horton Conway
- The Triangle Book, 
- The Symmetries of Things,