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Playful mathematician described as 'the most magical in the world'
John Horton Conway, the Liverpool-born mathematician, who has died aged 82 from complications related to Covid-19, was one of the most original minds of the 20th and 21st centuries; Sir Michael Atiyah described him as "the most magical mathematician in the world". His contributions included surreal numbers, Philosopher's Football and the Doomsday algorithm, but he was best known for 'Game of Life'.
Conway made contributions to almost every branch of pure mathematics, including group theory, number theory, algebra, geometric topology, theoretical physics, combinatorial game theory, the theory of knots and geometry. But he was best known for inventing "Game of Life", the world's first "cellular automaton" - a system in which rules are applied to cells and their neighbours in a grid. As well as providing hours of innocent fun for computer geeks, Game of Life has important scientific applications in simulating the rise, fall and alternations of communities of living organisms.
It was in 1970 when, at Conway's instigation, a group of friends at Cambridge joined him in manipulating two groups of coloured stones on squares on a grid - one colour representing live cells and the other, dead cells. The aim was to observe how, by moving them according to a few simple rules, they might evolve to simulate complex, lifelike systems in which unpredictable outcomes occur.
Conway tried a variety of different rules to determine how each "cell"or square would respond to what was happening in its neighbourhood of eight adjacent cells and eventually came up with the answer:
(1) any live cell with fewer than two live neighbours dies of loneliness;
(2) any live cell with more than three live neighbours dies of overcrowding;
(3) any live cell with two or three live neighbours lives;
(4) any dead cell with three live neighbours returns to life.
The "game" (a misnomer since there are no players and the rules determine everything that happens) gained popularity when Martin Gardner wrote about Conway's "fantastic solitaire pastime" in Scientific American.
With the development of computer capabilities, it went on to become one of the most popular programs in existence, producing startling but satisfying flourishes of pattern and motion on the computer screen. It was also incorporated into programs that explore the possibilities of simulating human cognition and the potentialities of artificial intelligence and self-reproducing robots.
His association with "Life", as it became known, was the cause of some annoyance for Conway who felt, with some justice, that many of his other contributions to mathematics were weightier. Yet even his more important work arose out of his enthusiasm for games.
By his own estimation his most important contribution was his discovery in the 1970s of "surreal numbers" - an entirely new class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. But his discovery was based on his analysis of winning strategies in the Chinese board game Go.
The author or co-author of more than 10 books including The Atlas of Finite Groups (1985), one of the most important books in group theory, Conway was co-author of Winning Ways for Your Mathematical Plays, a bestselling collection of mathematical games. His more light-hearted contributions included Conway's Piano Problem (what is the largest object that can be manoeuvred around a right-angled corner in a fixed-width corridor?); Sprouts, a join-the-dot paper and pencil game; Philosopher's Football (or "Phutball"), and Dots and Boxes, a popular diversion at the annual "math camps" which Conway attended after his move to Princeton in 1987.
"Every now and then I do a very interesting piece of math, though not as often as I used to," he explained. "Most of the time I'm thinking of trivial math. But when I hit the white hot stuff, it is fabulous."
John Horton Conway was born in Liverpool on Boxing Day 1937 to Cyril Horton Conway, who, after leaving school aged 14, made a living playing cards before becoming a chemistry laboratory assistant, and his wife Agnes.
Young John was interested in mathematics from an early age (at the age of four he could recite the powers of two) and from Liverpool's Holt High School for Boys he won a minor scholarship to read the subject at Gonville and Caius College, Cambridge.
There he became interested in games, spending hours playing backgammon, and set to work with conscious determination to transform himself from a painfully introverted adolescent to an extrovert.
After graduation in 1959 he undertook research in number theory supervised by Harold Davenport, who once observed that, given a problem to solve, Conway would "return with a very good solution to another problem".
After taking a PhD in 1964 Conway was appointed lecturer in Pure Mathematics at the university and was offered fellowships at Peterhouse and Sidney Sussex. Invited to dine at High Table in the former, he chose the latter, suspecting that the dinner invitation had arisen out of fear that his table manners might not pass muster. Later he returned to his old college, Caius, as a fellow.
Conway was awarded the Berwick Prize by the London Mathematical Society in 1971. In March 1981 he was elected a fellow of the Royal Society. Then in 1983 he was appointed Professor of Mathematics. Three years later he left Cambridge after accepting appointment to the John von Neumann Chair of Mathematics at Princeton. The prizes kept coming.
Bearded and portly, with playful eyes, Conway looked the part of the maths genius, wearing sandals in all weathers and a selection of T-shirts which, when teaching geometry, he would sometimes pull up to illustrate the mathematical notion of curvature. When elected a fellow of the American Academy of Arts and Sciences in 1992 he turned up at the ceremony in green shorts.
In her 2015 biography, Genius at Play: The Curious Mind of John Horton Conway, Siobhan Roberts observed that Conway could factor large numbers in his head and could recite π from memory "to 1,111+ digits": "He's been known to carry on his person a few decks of cards, dice, ropes, pennies, coat hangers, sometimes a Slinky, maybe a miniature bicycle, all props he deploys to extend his winning imagination." At Princeton he arranged a campus tour called "How to Stare at a Brick Wall".
Conway's mathematical ideas such as the "grand antiprism" and the "Monstrous Moonshine" conjecture (a symmetry group that lives in 196,883 dimensions) often contained an element of whimsy. His "Doomsday rule" was an algorithm to determine the day of the week for a given date. With a colleague, Simon Kochen, he developed the "Free Will Theorem", a mathematical formulation which claimed to prove that if humans have free will, so do elementary particles.
He was not, perhaps, the easiest man to live with. His first two marriages, to Eileen Howe and Larissa Queen, ended in divorce. One of them described her ex-husband to Siobhan Roberts as "the most selfish, childlike person I have ever met," adding: "One of the reasons I find that so intolerable is that I know damn well he can be human if he cares enough to bother."
He is survived by his third wife Diana, by their son, by four daughters from his first marriage (whose birth dates he remembered by classifying them as "the 60-Fibs", since they were born in 1960 plus the Fibonacci numbers), and two sons from his second marriage.
John Horton Conway, born December 26 1937, died April 11 2020
16 April 2020 © Telegraph