# Significant figures: John Conway

The story of an unforgettable mathematician.

Image: Thane Plambeck, CC BY 2.0 and Hyperdeath, CC BY-SA 3.0

A game of Sprouts starting with five dots—exactly one more move is possible.

My most memorable encounter with the work of late mathematician John Horton Conway came from a friend of mine I met as a first year graduate student. As we sat across from each other in the department common room, each having made little progress with our research, he slid me a piece of paper with five dots drawn on it. This game, he explained, consisted of us each taking turns to draw a line between any two dots, with the midpoint of the line we drew then counting as an additional dot. Although the lines could bend in any direction, they were not allowed to intersect each other, and each dot could join at most three line segments. The game was over when one player could not make any more moves, and the other player was declared the winner. At first, I was quickly defeated, and I spent quite some time trying to come up with the best strategies against my skilled opponent.

The game that we spent our lunchtime playing was Sprouts, invented by Conway and his friend Michael Paterson during their time at the University of Cambridge, and was later popularised by Martin Gardner in his Scientific American column Mathematical Games. Conway is perhaps best known for his interest in games: he invented many, and his two books on the subject On numbers and games and Winning ways for mathematical plays include detailed analyses of many two-player games. He was a regular contributor to Gardner’s column, and was a major figure in the world of recreational mathematics in his own right.

As a graduate student, Conway proved that every positive whole number can be written as the sum of at most 37 fifth powers.

Born in 1937, Conway grew up in Liverpool, and attended Cambridge as an undergraduate, staying on for his postdoctoral research in number theory, and eventual appointment as fellow and lecturer. He moved to Princeton in 1986, where he remained for the rest of his career. According to those who knew him, he was always ready to play: he would carry around puzzles, pennies, coat-hangers, and dice on him, ready to stoke the imagination of some unwitting colleague with a lively demonstration or challenge. Often described as charismatic, he certainly fulfilled certain stereotypes of the eccentric mathematician, but was also an inspiration for many of those he taught and spoke to, and remains so even after his death in April 2020 from complications due to Covid-19.

One of Conway’s most famous inventions was the game of life, a very simple type of ‘game’, that takes place on a grid of pixels. Each pixel starts either switched on or off, and each second, any off pixel with exactly three on neighbours will also switch on, and only on pixels with two or three on neighbours will remain on. Despite these simple rules, the game of life is actually so-called Turing complete, meaning that in theory, any computer programme could be run using these pixels. This is an example of a cellular automaton, for more check out this article

Conway was a prominent mathematician, not only dedicated to his work on popular games: on the contrary, his willingness to approach any topic with the same enthusiasm led to him contributing to research fields across mathematics. His interests included number theory, topology, analysis, group theory, classical geometry, even theoretical physics. Analysts, for example, may be familiar with his base 13 function, a function that takes every value between 0 and 1, but is discontinuous everywhere. Among academics, he is better known for his work in group theory, in particular on sporadic simple groups and the Monstrous Moonshine conjecture: a mathematical theory that connects the sporadic groups, mysterious algebraic structures coming from group theory, with functions called modular forms, coming from analysis. His name continues to be relevant, not only through his own considerable research, but also through those who took inspiration from him. In 2018, in a branch of topology called knot theory, a long-standing conjecture was solved by then-graduate student Lisa Piccirillo, which involved the classification of a knot which bears Conway’s name.

But for all his contributions, it was Conway’s willingness to collaborate, and share his love of ideas, that are an example for all those interested in mathematics. So, to live by that example, I encourage you to pick up some pencils and sheet of paper, find a friend, and go play a game of Sprouts.

Jamie is a second year mathematics student at Christ’s College Cambridge. His main interests are in group theory, number theory, and a touch of algebraic geometry.

Jakob is a PhD student and mathematician from London and works mainly in differential geometry. In his spare time, he likes to draw, and think about mathematics in art.

• ### In conversation with Shannon Trust

We find out more about the charity's work to support maths education in prisons.
• ### In conversation with Martin Hairer

We talk to the Fields medallist about his life, his work and his advice to his younger self.
• ### Page 3 model: The potato Earth

How Kelvin estimated the age of the Earth by thinking about a baked potato
• ### Page 3 model: Woollen jumpers

How did your favourite sweater become so big? And how big can it get?
• ### In conversation with Ulrike Tillmann

Sophie Maclean and David Sheard speak to a very top(olog)ical mathematician!
• ### On conditional probability: Cards, Covid, and Crazy Rich Asians

Madeleine Hall explores the sometimes counterintuitive consequences of conditional probability to our everyday lives.