The mathematical game player
A tribute to John Horton Conway 
(26 December 1937 --- 11 April 2020)
Many lives have been lost to COVID-19 in the last few months, but for the world of mathematics, one loss was especially sad, that of John Conway, a legend in his lifetime.
A distinct characteristic of Conway was that he loved to play games, preferably silly children's games, all day long. He would spend whole summers going from one mathematics camp to another, one for middle school children here and one for teenage students, and in all of them, he would play games with children, pose and solve puzzles. He would carry all sorts of things with him: decks of cards, dice, ropes, coins, coat hangers, sometimes a Slinky, even a miniature toy bicycle. These were all props he would use for explaining ideas, though Conway insisted that they more more for his own amusement. 
Unlike most other mathematicians, Conway was always obsessed with seemingly trivial concerns. He would be constantly factoring large numbers in his head.  He could recite more than a 1000 digits of pi from memory. He developed algorithms you could use to calculate the day of the week for any given date (in your head!), to count the number of steps while you climb stairs without actually counting, .... But importantly, Conway claimed that it was such thinking that led to his mathematical research, and his colleagues at Princeton agree.
If anyone has ever approached all mathematics in a spirit of play, it is Conway.  He also led the world in the mathematics of playing games. Take any two player board game like Chess. We can ask, does one of the players have a *winning strategy*: a way to play in such a way that no matter what moves the other player makes, she is assured of a win? If I have a strategy in game g and another in game h, how do I combine them in a game made of g and h as subgames? Such considerations lead to a beautiful algebra of games, and Conway made pioneering contributions to it.
Conway is credited with founding the area of Combinatorial Game Theory, a rich and beautiful subject of mathematical study. Conway also created many games for people to play.  In fact, Conway is best known to the public for designing the Game of life, a great boon to screen savers of computers.

The Game of life
This is a very simple game. Take a sheet of grid paper (graph paper or any paper with squares), and keep a pencil and eraser with you. Each square is called a cell and you can see that each has 8 neighbours.  In this game, every cell can be alive or dead at any instant. There are (only!) two rules:
A dead cell having exactly three live neighbours comes alive at the next instant (otherwise it stays dead).
An alive cell that has two or three alive neighbours, stays alive (else it dies).
Try out the evolution of the game starting from some initial configurations suggested in the Figures.
Questions: if you start with any configuration of alive and dead cells, can we predict whether we would keep getting new configurations, or settle down to a specific configurstion, or keep oscillating between some configurations? Pointing to a cell, can we figure out whether it will live for ever after some finite point in time? We can ask a variety of such questions.  It turns out that there is no uniform algorithm to answer any of these questions. In fact, the Game of Life is exactly as powerful as the digital computer in a theoretical sense, and there is a variety of questions of this kind linking the game to computation theory and complexity theory. 
Conway tinkered with the rules for years before he arrived at such simplicity. He did not use any computers during this search either, he hand calculated the evolution of many configurations. This was typical of Conway, the search for extreme simplicity encapsulating almost universal capability, and doing it all "in the head".

Conway was born in Liverpool, England and went to study in Cambridge on a scholarship. In the 1960s, he worked on sphere packing. Suppose that you want to fit as many circles as possible into a region of the Euclidean plane. How can one do this? Divide the plane into a grid made up of hexagons and circumscribe the largest possible circle inside each hexagon. The grid, called a hexagonal lattice, serves as an exact guide for the best way to pack circles in two-dimensional space. In three-dimensional space, we try and fit as many spheres as possible. In 4-dimensional space? We need 4-dimensional spheres!  In the 1960s, John Leech came up with a similar lattice for the most efficient packing of 24-dimensional spheres in 24-dimensional space.  Conway studied the symmetries of the Leech lattice, leading to what is called the Conway Group now. This led him to study the properties of similar groups, in even higher dimensions. 25 dimensions? Read on.
In a paper in 1979, Conway and Simon Norton conjectured (guessed a pattern of) a deep and surprising relationship between the properties of the so-called monster group and that of an object in number theory called the j-function. The paper was titled *Monstrous Moonshine*!  The monster group is a collection of symmetries that appear in 196,883-dimensional space.  In contrast, we live in just a 3-dimensional space! A decade later, Borcherds proved the conjecture, which won him the Fields medal in 1998.
Another area of mathematics in which Conway made an amazing contribution in was *knot theory*, a branch of topology. Knots can be thought of as closed loops of string. A fundamental problem in the area is that of knot equivalence: can one apply finitely many allowed operations to obtain one from another? Mathematicians have different kinds of tests they can apply that act as invariants: if applying them led to a pair of knots leads to different knots, the pair was different. One such test is called the Alexander polynomial, which is effective but not unique: the same knot could give rise to different Alexander polynomials. Conway fixed this, leading to what is known as a Conway polynomial, a fundamental tool in knot theory now. Another interesting contribution of Conway was an arrangement of knots, akin to the periodic table, that makes their properties easy to study.
Overall, Conway was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory, with some work also in geometry, geometric topology, algebra, analysis, algorithmics and theoretical physics. The vast range attests to Conway's ability to work on pretty much any mathematically posed question.  Conway was at Princeton University for the last quarter century, interacting intensively with students and colleagues. His biography by Siobhan Roberts is a deeply inspiring story.
Conway was known for his obsession for reducing proofs to the simplest terms. He gave 7 different proofs that the square root of 2 is irrational, analysing them to bring down the assumptions needed to the least possible. For John Conway, doing mathematics was always in a spirit of play; occasionally some theorems might come up for publication, but that is not the main aim.