Born in Liverpool, Conway was educated at Cambridge where he obtained his PhD in 1964. He was appointed professor of mathematics at Cambridge in 1983. In 1988 he moved to America to become John von Neumann Professor of Mathematics at Princeton.

Conway's name became familiar to a large number of nonmathematicians through his invention of the ‘game of life’ in 1970. The game is played on an infinite two-dimensional cellular array. Each cell has eight immediate neighbors, and can be in one of two states: on or off, occupied or empty, alive or dead. Two simple rules govern the outcome of any initial state:

1. A live cell will remain alive in the next generation if it has either two or three live neighbors.

2. An empty or dead cell will be occupied or come to life in the next generation if it has exactly three live neighbors. In all other situations living cells die and dead cells remain dead.

The game starts with an initial pattern of live cells and proceeds by a series of discrete changes – at each change all the cells change simultaneously to give a new pattern in the next generation.

Is there a general way, Conway asked, to determine the fate of any pattern? He also offered a $50 prize to anyone who could produce a pattern that grew indefinitely, or demonstrated that no such pattern could exist. The problem so intrigued computer operators that the search for interesting Life forms is thought to have cost millions of dollars in unauthorized computer time. The prize was awarded to a group at the Massachusetts Institute of Technology which discovered a pattern, a ‘glider’, which every thirty generations produced another glider. Conway used gliders to demonstrate in 1982 that there are patterns that behave like self-replicating animals. The snag is that spontaneous creations of such patterns would require a computer screen larger than the solar system.

Conway has also made major contributions to group theory and knot theory. Is it possible to classify all finite simple groups? These are groups which, in the manner of prime numbers, cannot be decomposed into smaller groups. While many groups fitted into clearly defined classes, several others, known as sporadic groups, fitted into no recognized class. Five such groups were identified by Mathieu in the 1860s. A sixth was discovered in 1965 and Conway identified a further three in 1968. By 1975 twenty-six sporadic groups were known. This completed the classification theorem, also called the ‘Enormous Theorem’, which classifies all finite simple groups and has been estimated to be 15,000 pages long.

One aim of knot theorists is to distinguish between different types of knots. This can be done by calculating the crossing number, that is, the number of points the string crosses itself. Unfortunately, many different knots can have the same crossing number, and the number itself may be difficult to calculate. In 1960 Conway introduced a new and simpler way to find crossing numbers. It allowed him to establish, for example, that there are at least 801 distinct knots with a crossing number no higher than 11.

Conway is also the author, along with E. Burkelamp and R. Guy, of a book on mathematics and games – Winning Ways (1982).