John Horton Conway, an English-born Princeton mathematician whose body of work ranged from the rigorously highbrow to the frivolously fun, earning him prizes and a reputation as a creative, iconoclastic, and even magical genius, died Saturday in New Brunswick, N.J. He was 82.

His wife, Diana Conway, said his death, at a nursing home, was caused by COVID-19.

John Conway’s boundless curiosity produced profound contributions to number theory, game theory, coding theory, group theory, knot theory, topology, probability theory, algebra, analysis, combinatorics, and more. Foremost, he considered himself a classical geometer.

“His swath was probably broader than anyone who ever lived,” said mathematician Neil Sloane, a collaborator with Mr. Conway and the founder of the On-Line Encyclopedia of Integer Sequences. “I’ve worked with a lot of people, and he was the fastest at solving a problem and would pursue a topic as far as it would go.” (The two were coauthors of 50 papers and published the 706-page book “Sphere Packings, Lattices and Groups.”)

During what Mr. Conway called his “annus mirabilis,” roughly 1969 to 1970, he discovered what’s known as the Conway group, an entity in the realm of mathematical symmetry that inhabits 24-dimensional space. He discovered a new type of number, “surreal numbers.” And he invented the cellular automaton Game of Life, which is among the most beautiful mathematical models of computation. He described it as a “no-player never-ending” game.

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His friend Martin Gardner, the longtime mathematical games columnist for Scientific American, called the Game of Life Mr. Conway’s “most famous brainchild.” He reckoned that at the game’s peak of popularity — with users programming it at home and at work — one quarter of the world’s computers were playing it.

“Conway’s LIFE changed mine,” musician Brian Eno said in an e-mail. “I think Conway himself thought it rather trivial, but for a nonmathematician like me, it was a shock to the intuition, a shattering revelation — to watch glorious complexity emerging from staid simplicity.”

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Mr. Conway was proudest of his discovery of surreal numbers. (The Stanford computer scientist Donald Knuth had come up with the name while writing the novelette “Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.”)

Described by Gardner as “an astonishing feat of legerdemain,” the surreals are a super-continuum of numbers, including all the old-fashioned real ones (integers, fractions and irrationals like pi) as well as those that go above, beyond, below and within, embracing both the infinites and the infinitesimals.

Mr. Conway always hoped that surreal numbers might find practical applications, perhaps in helping to illuminate the universe on the cosmic and quantum scales.

One of Mr. Conway’s favorite accomplishments was the Free Will Theorem, conceptualized casually over the course of a decade with his friend and fellow Princeton mathematician Simon Kochen and first published in 2006 (and later revised).

The theorem, simply put, is this: If physicists have free will while performing experiments, then elementary particles possess free will as well. And this, Mr. Conway and Kochen reckoned, probably explains why and how humans have free will in the first place.

“In mathematics and physics there are two kinds of geniuses,” Kochen said by phone from his home in Princeton, echoing something once said about the physicist Richard Feynman. “There are the ordinary geniuses — they are just like you and me but they are better at it; if we’d worked hard enough, maybe we could get some of the same results.

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“But then there are the magical geniuses,” he added. “Richard Feynman was a magical genius. And the same always struck me about John — he was a magical mathematician. He was a magical genius rather than an ordinary genius.”

John Horton Conway was born Dec. 26, 1937, in Liverpool, England, the third child and only son of Cyril and Agnes (Boyce) Conway. His father, an autodidact, had left school at age 14 and, with his photographic memory, made a living playing cards. Later he was a technician in the chemistry lab at the Liverpool Institute High School for Boys, setting up experiments for students, among them George Harrison and Paul McCartney.

Mr. Conway’s mother, a great reader, especially of Dickens, had worked from age 11. Family lore has it that she boasted about finding her son at age of 4 reciting the powers of two. At 18, in 1956, Mr. Conway left home for the University of Cambridge, where he earned his doctorate. His adviser, Harold Davenport, a number theorist, once said that when he would give Mr. Conway a problem to solve, “he would return with a very good solution to another problem.”

As a student, Mr. Conway cultivated his acknowledged lifelong preference for being lazy, playing games and doing no work. He could be easily distracted by what he called “nerdish delights.” He once went on a flexagon binge, courtesy of Gardner, who described flexagons as “polygons, folded from straight or crooked strips of paper, which have the fascinating property of changing their faces when they are flexed.”

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He built a water-powered computer, which he called Winnie (Water Initiated Nonchalantly Numerical Integrating Engine). He read and annotated H.S.M. Coxeter’s edition of W.W. Rouse Ball’s classic work, “Mathematical Recreations and Essays” and wrote Coxeter a lengthy letter that started a lifelong friendship between these two classical geometers.

Hired at Cambridge as an assistant lecturer, Mr. Conway gained a reputation for his high jinks (not to mention his disheveled appearance). Lecturing on symmetry and the platonic solids, he might bring in a turnip as a prop, carving it one slice at a time into, say, an icosahedron, with its 20 triangular faces, eating the scraps as he went. “He was by far the most charismatic lecturer in the faculty,” his Cambridge colleague Peter Swinnerton-Dyer once said.

Mr. Conway invented a profusion of games — like Phutball (short for Philosopher’s Football, which is a little like checkers on a Go board) and collected them in the book “Winning Ways for Your Mathematical Plays,” in collaboration with Elwyn Berlekamp and Richard Guy.

In 1985, with four coauthors, he published “The ATLAS of Finite Groups,” one of the most important books in group theory.

That same year, he was invited to give a talk at Princeton, and a job offer followed: In 1987, he took up the position of the John von Neumann professor of applied and computational mathematics. In announcing the hire, Princeton’s president called Mr. Conway “one of the most eminent mathematicians of the century.”

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At Princeton, Mr. Conway, with his mischievous and seductive aura, drew news media attention. Asked by a reporter for The New York Times about his life of the mind, he replied: “What happens most of the time is nothing. You just can’t have ideas often.”

He became a fellow of the American Academy of Arts and Sciences in 1992. A fellow inductee, the mathematician Robert MacPherson, recalled that at the ceremony Mr. Conway accepted his honor in what appeared to be green running shorts.

His first two marriages, to Eileen Howe and Larissa Queen, ended in divorce.

In addition to his wife, he is survived by four daughters from his first marriage, Annie, Ellie and Susie Conway and Rosie Wayman; two sons from his second marriage, Oliver and Alex; a son with Diana Conway, Gareth; three grandchildren; and six great-grandchildren.

Math, Mr. Conway believed, should be fun. “He often thought that the math we were teaching was too serious,” said Mira Bernstein, a mathematician and a former executive director of Canada/USA Mathcamp, an international summer program for high school students. “And he didn’t mean that we should be teaching them silly math — to him, fun was deep. But he wanted to make sure that the playfulness was always, always there.”

Mr. Conway persevered in finding the fun through triple bypass surgery, a suicide attempt, and a number of strokes. Sometimes he would regale anyone willing to listen on the science of rainbows or on his Doomsday rule for calculating the day of the week for any given date.

And there were ever more games of Phutball, which Mr. Conway was not very good at. Sometimes, when all seemed lost — when he was almost certainly beaten at his own game, though he might yet magically prevail — he’d delight in borrowing from Mark Twain, admonishing his opponents, “Reports of my death have been greatly exaggerated!”