He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life.

[4] 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".

[7][8] 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 became interested in infinite ordinals.

[13] For instance, he discussed Conway's game of Sprouts (July 1967), Hackenbush (January 1972), and his angel and devil problem (February 1974).

Since Conway's game was popularized by Martin Gardner in Scientific American in 1970,[23] it has spawned hundreds of computer programs, web sites, and articles.

[25] 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.

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 developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conway's soldiers.

In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms.

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

Based on a 1978 observation by mathematician John McKay, Conway and Norton formulated the complex of conjectures known as monstrous moonshine.

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.

While lecturing on the 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".

[39] 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.

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.

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.