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Sprouts and the game of life: John Horton Conway 1937 - 2020

by Trevor Moore


Conway invented or helped invent much stuff to keep kids and mathematicians amused. Flexagons and Sprouts will keep kids amused for hours.

Some years ago, on a rainy afternoon in the UK, I was taking care of a bunch of bored youngsters. Eventually I said, “let’s play battleships”. “We can’t,” came the reply, “we haven’t got a set.” It was my introduction to the idea that a game needed a kit; it didn’t seem that you could play a game without one. With the game of Sprouts, however, it’s difficult to see what a set or a kit would comprise of. To play it you need a piece of paper and at least one pencil. Having written that, if you were on the beach you could play it with your fingers in the sand.

The game is played by two players, who start with a few (four is a good number) spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line between two spots (or from a spot to itself) and adding a new spot somewhere along the line.

There are some rules:

1. The line may be straight or curved but must not touch or cross itself or any other line.

2. A new spot must split the line into two shorter lines; you cannot put a spot on top of another one

3. You cannot have more than three lines attached to a spot. Note that a line from a spot to itself counts as two attached lines and new spots are counted as having two lines already attached to them.

There are two ways to compete: in “normal” play the player who makes the last move wins, in “misère” play the player who makes the last move loses. It’s up to you which you do and if you don’t have a friend you can play it with yourself though it’s not as much fun.

Above: A 2-spot game of Sprouts. The "normal" game ends when the first player is unable to draw a connecting line between the only two free points, marked in green.

The inventor of this game was a man called John Horton Conway. I say “was” because he died on 11 April 2020 of COVID-19 at the age of 82. The world of mathematics will be a sadder, if perhaps a bit less whacky place, as a result of his death. He was a truly inspirational mathematician. One of my favourite papers of his is one called The Free Will Theorem. My interpretation of the result that he proves (with co-author Simon Kochen) is that human beings have free will if and only if quarks have indeterminate spin. Now we know from the great physicist Werner Karl Heisenberg that these sub-atomic particles do indeed have a spin that is indeterminate. What this means, of course, is that there is no Grand Plan for the universe; it’s all up to us.

Conway is also known for The Game of Life though eventually he did not want to be reminded about it. I suppose it’s a bit like the Rolling Stones being asked to play Satisfaction for the 10,000,000th time. Conway would heckle people at conferences who made the mistake of mentioning The Game of Life.

John Horton Conway, 1932 - 2020

Now, you certainly can play The Game of Life without a kit. It was invented on a coffee table in a staff common room at a University. And even if Conway became bored with it, you will almost certainly enjoy it. The Game of Life is what is called a cellular automaton because it works with a grid of cells and some rules about what might happen to the contents of those cells. You start with a grid and you mark some of the cells with an “X” (or anything you like). A cell with an “X” in it is alive; a cell without is dead. Then there are some rules:

1. Any live cell with fewer than two live neighbours dies, as if by underpopulation

2. Any live cell with two or three live neighbours lives on to the next generation

3. Any live cell with more than three live neighbours dies, as if by overpopulation

4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction

The best way to see what happens is to search for a Game of Life program on the Internet. You can easily while away an hour or two experimenting. Conway described it as a “no-player never-ending” game.

VIDEO:Inventing Game of Life (John Conway) - Numberphile

Conway didn’t spend his life playing games although he wrote a well-known (well, well-known if you are into that sort of thing) 1975 book On Numbers and Games. But that book tells one something about Conway’s approach to life. The first chapter is Chapter 0 and at its start Conway provides a quote: “whatever is not forbidden is permitted”. His biographer, Siobhan Roberts, notes that “he made profound contributions to number theory, coding theory, probability theory, topology, algebra and more — and created games from it all.”

He was born on Boxing Day 1936 to a father who made a living playing cards and who later became a chemistry lab technician at the Liverpool Institute High School for Boys and set up experiments for George Harrison and Paul McCartney among other students.

As a student at Cambridge, Conway cultivated a lifelong preference for being idle. He had worried about being idle but then realised that idleness often leads to ideas and he was all about ideas. He stayed at Cambridge and became a professor of mathematics and a Fellow of the Royal Society in 1981. In 1987 he took up the position of the John von Neumann professor of applied and computational mathematics - probably one of the top 3 or 4 mathematics chairs in the world. It was in Princeton that he died.

A hexaflexagon, shown with the same face in two configurations

Although he made contributions to so many areas of mathematics, I think it was his playfulness that always bubbled to the top; hence the 1975 book On Numbers and Games. Conway was fairly close to a recreational mathematician called Martin Gardner. He used to write for Scientific American and my shelves are graced with several of his books. Conway once went on a flexagon binge, courtesy of Gardner, who described flexagons as “polygons, folded from straight or crooked strips of paper, which have the fascinating property of changing their faces when they are flexed.” You can look up flexagons on Wikipedia or at The Flexagon Portal. These things will provide any child with yards of fun. Then you can move on to Phutball but I will leave it to you to look that one up.

Further reading:

Genius at play: The curious mind of John Horton Conway, Siobhan Roberts, 2015, ISBN 978-1-62040-593-2, 454pp

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