John Horton Conway was a British mathematician who made fundamental contributions to group theory, number theory, knot theory, and combinatorial game theory; he discovered the Game of Life and surreal numbers. Throughout his career he held professorships at Cambridge and Princeton universities and conducted pioneering work on the classification of finite simple groups and sphere packings.

John Horton Conway was born on 26 December 1937 in Liverpool, England, to Agnes Boyce and Cyril Horton Conway. His interest in mathematics began at an early age; it is recorded that by the age of four he could recite powers of two from memory and, during primary school, could calculate the day of the week for any given date. At the age of eleven, during his secondary school entrance interview, he stated his intention to become a mathematician at the University of Cambridge. After completing secondary education, he entered Gonville and Caius College at Cambridge and received his BA degree in 1959. Conway pursued his graduate studies in number theory under Harold Davenport, proving that every integer can be expressed as the sum of 37 fifth powers, thereby contributing to Waring’s Problem, and earned his doctorate in 1964.^【1】

After receiving his doctorate, Conway was appointed as a lecturer in Pure Mathematics at Cambridge University in 1964 and was elected a research fellow at Sidney Sussex College the same year. During this phase of his career he initially worked on mathematical logic; however, in the mid-1960s he turned his attention to a problem posed by John Leech concerning 24-dimensional sphere packings, known as the Leech lattice. In 1968, by analyzing the symmetry group of the Leech lattice, he discovered previously unknown finite simple groups now called the Conway groups. This work represented a major step in the classification program for finite simple groups and formed the foundation of the Atlas of Finite Groups, a 15-year project published in 1986. He also investigated with Simon Norton the connection between the Monster group, also known as the Friendly Giant, and modular functions, known as the Moonshine conjecture.^【2】

The period 1969–1970, which Conway referred to as his annus mirabilis, saw the emergence of his three most famous contributions: the Game of Life, surreal numbers, and the Conway groups. The Game of Life, announced in 1970 in Martin Gardner’s column in Scientific American, sparked extensive research in cellular automata and was later proven capable of simulating a Turing machine.^【3】 The surreal numbers, derived from his analysis of the game of Go, provide a systematic structure encompassing both infinitesimals and infinite numbers. In recognition of these academic achievements, Conway was promoted to reader, a senior lectureship, at Cambridge in 1973 and appointed professor of mathematics in 1983.

Conway’s research spanned a broad range from knot theory to theoretical physics. In knot theory, he developed a new classification system for knots and discovered the Alexander-Conway polynomial. In geometry, in collaboration with Neil Sloane, he co-authored Sphere Packings, Lattices and Groups, published in 1988, which addressed high-dimensional sphere packings applied to error-correcting codes. In 2004, with Simon Kochen, he formulated the Free Will Theorem, which examines the relationship between an experimenter’s choices and the behavior of particles within the framework of quantum mechanics.^【4】 He also, together with Michael Guy, identified the 64 convex uniform polyhedra.

In 1986, Conway left Cambridge to assume the John von Neumann Chair of Applied and Computational Mathematics at Princeton University. During his tenure at Princeton he actively taught foundational courses such as Real Analysis in the mathematics department and participated in summer camps for young talents. Beyond his academic research, he developed numerous mathematical games and puzzles, including Sprouts, Phutball, and Conway’s Soldiers. Known for his informal office environment and for constructing physical models from paper and wood to illustrate mathematical concepts, Conway became a central figure in the social and academic life of the department.

John Horton Conway was married three times and had seven children: Susan, Rose, Elena, Ann-Louise, Alex, Oliver, and Gareth. He married Diana in 2001 and lived in Princeton, New Jersey, until his death. Throughout his career he received numerous honors including the Berwick Prize of the London Mathematical Society, fellowship in the Royal Society, the Pólya Prize, the Nemmers Prize in Mathematics, and the Leroy P. Steele Prize for Mathematical Exposition. Conway died in Princeton on 11 April 2020.