Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Game of Life

      Maths in a minute: Conway's Game of Life

      Chris Budd
      2 December, 2024

      John Conway's Game of Life is a cellular automaton, which is supposed to produce patterns that resemble living organisms. It was developed by the mathematician John Conway in the 1970s.

      The Game of Life is played on an infinite two-dimensional grid of square cells.  A cell that is green is considered to be alive and a cell that is grey is considered to be dead. Every cell interacts with its eight neighbours which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

      1. Any live cell with fewer than two live neighbours dies (as if caused 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 initial pattern constitutes the seed of the system. The first generation is created by applying the above rules, simultaneously, to every cell in the seed. As with the one-dimensional cellular automata, the rules continue to be applied repeatedly to create further generations. Depending on the seed, very exotic patterns can emerge.

      You can explore the Game of Life in the interactivity below, created by our amazing colleague Oscar Gillespie. It starts off with a random pattern of live cells. If you would instead like to choose your own starting arrangement of live cells, click Clear the grid and then click on the individual cells you would like to be live. To run just a single step of the Game of Life, click on Run a generation. To run through many steps, click Run. And to start again, click Stop and then Reset.



       

      Conway's game of life shows that even very simple rules can produce a large variety of different behaviours, including highly organised and very complex ones. What's amazing is this wasn't what Conway set out to show when he designed the game — what he was really after was an infinitely programmable computer. Find out more in our interview with Conway.

      Here are some interesting initial patterns to play with:

       

      • The pulsar

         


         

      • A tetromino

         


         

      • A diehard

         


         

      • A glider

         

      • A gun

         

       


      About this article

      This article is based on an extract from Cellular automata by Chris Budd. The interactivity was produced by Oscar Gillespie.

      • Log in or register to post comments

      You might also like

      article

      Cellular automata

      Find out how a square grid and some simple rules can generate complex patterns and life-like behaviour.

      article

      Maths in a minute: Cellular automata

      Mindless games can produce surprising results.
      article
      aliens

      Games, Life and the Game of Life

      When we finally meet the Martians, John Conway believes they are going to want to talk mathematics. He talks to Plus about his Life game, artificial life and what we will have in common with extraterrestrials.

      Read more about...

      game of Life
      cellular automaton
      dynamical system
      Maths in a minute
      University of Cambridge logo

      Plus is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms