Conway’s Game of Life

by Scott Kim

For my July 15th presentation to the online Math Teacher Circle Workshop, I chose to introduce teachers to Conway’s Game of Life, a deeply fascinating game with deceptively simple rules. I chose Life because I think it is one of the most thought-provoking yet accessible mathematical creations of the last century.

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The rock star mathematician John Conway (he was born in Liverpool) was best known for his deep exploration of games. The Game of Life is Conway’s best-known game. Life (not to be confused with the Hasbro board game) is not a game in the usual sense of a competition between two players. It is more like a simulated world where you set up the primordial soup, and then let life evolve.

Imagine an infinite checkerboard. Each square can be empty, or have a checker on it. You set up the initial pattern of checkers. Then the rules of Life take over, and dictate how your pattern grows and changes from one state to the next. The rules are simple, but the behaviors that emerge are anything but. Some patterns die out, others settle into a repeating cycle, and others, like this example, fly across the board like illustration, dubbed a medium-weight spaceship.

How to play Conway's Game of Life

BOARD. Life is played on an infinite grid of squares, called “cells”. Each cell has 8 neighbors — the adjacent cells it touches at an edge or corner.

PIECES. Each cell can have a piece in it or be empty.

SETUP. At the beginning of the game we clear the board, then put pieces in some of the cells. This pattern is generation 1 of our Life game. 

MOVES. Once generation 1 is set, the game makes moves automatically by itself. Pieces live, die or are born from one generation to the next, by these rules:

  1. DEATH. A piece with 1 or fewer neighbors dies of loneliness. A piece with 4 or more neighbors dies of overcrowding.
  2. STAYING ALIVE. A piece with 2 or 3 neighbors stays alive to the next generation.
  3. BIRTH. An empty cell with exactly 3 living neighbors gives birth to a new piece. Other empty cells stay empty.

GOAL. See what happens to your pattern!

Each Cell has 8 neighbors

Here is generation 1 of a Life game. I’ve marked the pieces that will die with X, and the cells that will give birth with a heart.

Here is generation 2, after pieces from generation 1 have died and been born

When Life first appeared in the pages of Scientific American magazine in 1970, it made quite a splash. This was not mathematics as usual. Programmers all over the world coded up their own life apps, and soon more computer time had been devoted to exploring Life than any other endeavor on the planet. 

Life is a deep game for the same reason that computers are a deep invention. To study Life is to study the limits of what is computable. The black and white squares of a Life game are like the bits in a computer, and the rules of Life are like the wiring of the CPU. In fact, Life is a particular case of a cellular automaton — the theoretical model explored by the people like Alan Turing in the first half of the 20th century who invented the first electronic computers. 

Playing Life on the web

Beginner Life engine ( for small patterns): https://academo.org/demos/conways-game-of-life/

Expert Life engine (Golly, for huge patterns and alternate rule sets): https://sourceforge.net/projects/golly/

Try using the above links to explore the fate of these patterns:

4 in a row — honeycomb (still life)

5 in a row — oscillator, period 2

6 in a row — ??????

7 in a row — still life: 4 honeycombs

8 in a row — still life: 4 honeycombs, 4 blocks

9 in a row — ??????

10 in a row — oscillator, period 15

You can read about the history of Life here, and listen to Conway talk about Life here.

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