Conway's Game of Life — Can Mathematics Become Alive?
Four simple rules. An infinite chessboard. And behaviour nobody explicitly programmed.
Imagine drawing a few squares on a sheet of graph paper, applying four very simple mathematical rules, and watching the pattern apparently come alive.
Some shapes disappear almost immediately. Others remain unchanged indefinitely. Some alternate between different arrangements, while a few seem to travel across the paper as though they have somewhere to go.
Even more remarkably, certain patterns can produce an endless stream of moving structures.
There is no intelligence controlling them. No computer player is making decisions. Nobody has programmed the shapes to move, reproduce or interact.
Everything follows from four elementary rules.
Welcome to Conway's Game of Life, one of the most fascinating examples of how simple mathematics can produce extraordinary complexity.
It is also a wonderful way to introduce GCSE and A-level students to mathematics they are unlikely to encounter in their normal lessons.
1. A Mathematical Game Without Players
The Game of Life was developed by the British mathematician John Horton Conway and became widely known in 1970 through Martin Gardner's Mathematical Games column in Scientific American.
Despite its name, it is not really a conventional game.
There is no opponent, no score and no winning move.
Instead, it is what mathematicians call a cellular automaton.
That sounds complicated, but the underlying idea is remarkably straightforward.
Imagine a sheet of squared paper extending indefinitely in every direction.
Each square, or cell, has only two possible states:
Alive — represented by a filled or coloured square.
Dead — represented by an empty square.
Time advances in discrete steps called generations.
At each generation, every cell examines the eight squares surrounding it and changes according to the same rules.
The fascinating part is that the cells do not need to know anything about the rest of the grid.
They only respond to their immediate neighbours.
Yet when thousands of cells follow those rules simultaneously, remarkable structures emerge.
2. The Four Rules That Govern Life
Each cell has eight possible neighbours: above, below, left, right and the four diagonals.
The four rules are:
Rule 1: Underpopulation
A living cell with fewer than two living neighbours dies.
Rule 2: Survival
A living cell with two or three living neighbours survives into the next generation.
Rule 3: Overpopulation
A living cell with more than three living neighbours dies.
Rule 4: Reproduction
A dead cell with exactly three living neighbours becomes alive.
That is the entire mathematical system.
One particularly important detail is that all cells update simultaneously. We calculate the next generation from the existing one, rather than changing squares individually as we examine them.
Otherwise, the order in which we visit the cells would influence the result.
The rules are often abbreviated to B3/S23.
B3 means that a dead cell is born when it has exactly three living neighbours.
S23 means that a living cell survives when it has two or three living neighbours.
Nothing in these rules explicitly says that a structure should move, oscillate, grow indefinitely or behave like a miniature machine.
Nevertheless, all these things are possible.
3. Our First Experiment — Three Squares in a Row
We do not need a computer to start exploring.
Take some squared paper and colour three adjacent squares in a horizontal line.
Represent living cells by X and dead cells by a dot.
Generation 0
.....
.....
.XXX.
.....
.....Apply the four rules to every square.
The cells at either end of the line have only one living neighbour, so they die.
The centre cell has two living neighbours and survives.
Meanwhile, the empty squares immediately above and below the centre each have exactly three living neighbours.
Both become alive.
Generation 1
.....
..X..
..X..
..X..
.....Our horizontal line has become vertical!
Now apply the rules again.
Generation 2
.....
.....
.XXX.
.....
.....We are back where we started.
The pattern will alternate between these two arrangements indefinitely, provided nothing else interferes.
This little structure is called a blinker.
It is an example of an oscillator, a mathematical pattern that repeats after a fixed number of generations.
The blinker has a period of two.
Period = number of generations required for a pattern to repeat.
Notice something else.
In every generation, the blinker has exactly three living cells.
Its position and total population remain unchanged, but its shape changes.
This gives students three different mathematical quantities to investigate: population, position and repetition period.
4. When Nothing Changes — Still Lifes
Not all patterns move or oscillate.
Try starting with four living squares arranged into a 2 × 2 block.
....
.XX.
.XX.
....Each living cell has exactly three living neighbours.
Therefore, every cell survives.
No surrounding dead cell has exactly three living neighbours.
Consequently, no new cells appear.
The pattern remains unchanged forever.
This is called a still life.
It provides an excellent opportunity to ask students a question:
Can you design a different pattern that remains completely unchanged?
One example is the beehive:
......
..XX..
.X..X.
..XX..
......Students can investigate whether every living cell has two or three living neighbours and whether any neighbouring dead cell has exactly three.
This turns an apparently recreational activity into an exercise in logical reasoning and proof.
Finding a stable pattern is one thing.
Proving that it is stable is rather more interesting.
5. The Glider — A Pattern That Travels
Now for something genuinely surprising.
Arrange five living cells as follows:
.X.
..X
XXXThis is called a glider.
Apply the rules four times.
The five-cell pattern will return to its original shape, but it will have moved one square diagonally downwards and to the right.
Continue the simulation and it will keep travelling.
The pattern does not literally contain the same five living cells throughout its journey. Cells die and new cells are born.
What moves is the arrangement of living cells.
This is rather like a wave travelling across water.
The water does not have to move along with the wave, but the pattern of disturbance does.
That distinction between the movement of matter and the movement of a pattern is a valuable idea in physics as well as mathematics.
How fast does a glider move?
The glider moves one square diagonally every four generations.
Its displacement per generation is therefore:
Horizontal displacement = 1/4 cell per generation.
Vertical displacement = 1/4 cell per generation.
Students familiar with Pythagoras' theorem can go further.
Diagonal displacement after four generations = sqrt(1² + 1²) = sqrt(2) cells.
Average diagonal speed = sqrt(2)/4, or approximately 0.354 cell widths per generation.
Strictly speaking, the glider's position is changing in discrete steps, not travelling continuously.
Nevertheless, we can use familiar ideas of displacement, direction and average speed to describe its behaviour.
This is a lovely example of GCSE mathematics appearing in an unexpected setting.
6. A Pattern That Produces Other Patterns
The glider is remarkable enough.
However, mathematicians discovered something even more extraordinary.
In 1970, Bill Gosper and colleagues at the Massachusetts Institute of Technology discovered a configuration now known as the Gosper glider gun.
It contains a structure that repeatedly produces gliders.
Every 30 generations, another glider emerges and travels away.
The original structure continues operating.
Given an infinite grid, an undisturbed glider gun can create an unlimited number of gliders over time.
That was mathematically significant because it demonstrated that a finite starting configuration could produce unbounded growth.
Conway had originally conjectured that this might be impossible.
The glider gun disproved that conjecture.
What I particularly like about this example is the way it illustrates the importance of experimentation in mathematics.
A mathematician can develop a hypothesis, explore examples, and sometimes discover a counterexample that overturns the original idea.
That is genuine mathematical investigation.
7. What Happens If We Start With Random Cells?
Instead of carefully designing a blinker or a glider, suppose we fill a large grid randomly.
Perhaps each square has a 30% probability of starting alive.
Initially, the pattern looks like scattered noise.
As generations pass, clusters form, collide and disappear.
Some settle into stable blocks.
Others become oscillators.
Occasionally, moving gliders escape from the surrounding activity.
What began as randomness can produce recognisable structure.
This is an example of emergent behaviour.
Emergence describes situations in which the interactions of relatively simple components produce complex large-scale behaviour.
We can find related ideas in many other fields:
Flocks of birds, where individuals respond to nearby birds.
Traffic flow, where simple driver interactions can produce congestion.
Ant colonies, where local interactions contribute to organised activity.
Crystal growth, where local physical processes generate large-scale patterns.
Computer networks, where many independently operating devices form a larger system.
These are analogies, not examples of systems following Conway's exact rules.
Nevertheless, they raise an important question.
How much complexity can arise from simple local interactions?
For students who have been taught to associate mathematics primarily with equations and calculations, this can be a rather different way of thinking.
8. Can Mathematics Really Become Alive?
The title raises an interesting question.
Have we actually created life?
Not in the biological sense.
A glider does not have cells containing DNA. It does not metabolise, maintain a biological internal environment or evolve by natural selection merely because it moves.
The Game of Life is a mathematical model, not evidence that a computer simulation has become biologically alive or conscious.
Nevertheless, it demonstrates something important.
Apparently purposeful behaviour does not always require a central controller.
A glider looks as though it is travelling towards a destination.
A glider gun looks as though it is manufacturing objects.
Some elaborate configurations can even transmit information and perform computations.
Yet every event is determined by exactly the same four rules.
This brings us to a deeper question.
When we see complicated behaviour, how can we determine whether it arises from intelligence, design or simple underlying processes?
That is a question extending far beyond GCSE mathematics.
9. From Coloured Squares to Computers
One of the most extraordinary properties of Conway's Game of Life is that it is capable of universal computation.
With sufficiently elaborate configurations, its patterns can represent information and implement logical operations.
In principle, arrangements of cells can simulate a general-purpose computer, given enough time and space.
The idea is not that every glider is a little computer.
Rather, carefully constructed interactions between patterns can be used to implement computational mechanisms.
This connects Conway's creation with the work of Alan Turing and the foundations of computer science.
It also explains why the Game of Life appeals to students studying A-level Computer Science alongside Mathematics or Further Mathematics.
The connection is much deeper than simply writing a program to draw coloured squares.
The evolving squares themselves can be organised to perform computations.
The same mathematical world that produces a three-cell blinker can also support remarkably sophisticated logical structures.
10. A Classroom Investigation — Can You Predict the Future?
Here is an investigation I would enjoy giving to a student.
Start with the following L-shaped arrangement:
.....
.XX..
.X...
.....
.....Without using a computer, predict the next generation.
Then predict the generation after that.
For the first generation, the correct result is a 2 × 2 block:
.....
.XX..
.XX..
.....
.....The pattern then remains stable.
This is a useful exercise because it demonstrates that three living cells do not necessarily behave in the same way simply because the total population is identical.
Their arrangement matters.
Extension question: Can you find two different initial configurations with the same number of living cells that behave completely differently?
Try a horizontal line of three living cells and the L-shaped arrangement above.
One becomes a repeating oscillator.
The other becomes stable.
Same number of initial cells. Same rules. Different outcomes.
That is an important lesson about initial conditions and structure.
11. Building Your Own Game of Life in Excel
Students do not need sophisticated programming software to explore cellular automata.
Excel or another spreadsheet can be used.
A possible approach is to:
Create a 20 × 20 grid with living cells represented by 1 and dead cells by 0.
Calculate the number of living neighbours surrounding each cell.
Apply the Game of Life rules to calculate the next generation on a separate worksheet.
Use conditional formatting to colour living cells.
Repeat the calculation to observe how the pattern evolves.
For example, if a cell is located at C3 and its neighbours occupy the surrounding 3 × 3 range, the number of living neighbours can be calculated using:
=SUM(B2:D4)-C3
Subtracting C3 removes the central cell from the count.
The next generation must be calculated separately using the original values for all cells.
For an A-level student, the spreadsheet opens the door to investigating populations, proportions, repeated patterns and mathematical modelling.
There is an important practical detail: Excel grids are finite.
A limited grid needs clearly defined boundary conditions.
We might treat every cell outside the grid as dead, or construct a wrapping grid in which the left edge connects to the right and the top connects to the bottom.
These choices can change the behaviour of a pattern.
In other words, even the boundary conditions form part of the mathematical model.
12. An Even Better Challenge — Write the Program Yourself
For students studying Computer Science, writing a Game of Life simulator is a rewarding programming challenge.
Python is an obvious choice, although JavaScript, C# and many other languages would work.
The central algorithm is:
Create a grid
Choose which cells begin alive
Repeat:
For each cell:
Count its eight living neighbours
If the cell is alive:
Keep it alive if the count is 2 or 3
Otherwise make it dead
If the cell is dead:
Make it alive if the count is 3
Otherwise keep it dead
Replace the old grid with the new grid
Display the resultNotice that we must calculate the complete new grid before replacing the old one.
A common programming mistake is to change cells directly while counting their neighbours.
That introduces an unintended dependence on the order in which the cells are processed.
The challenge therefore teaches more than loops and conditional statements.
It introduces algorithms, arrays, state, simultaneous updating, debugging and software testing.
A particularly good test is to check whether the program correctly reproduces the blinker.
If it does not return to its original position after two generations, something is wrong.
13. A-Level Extension — Mathematics Behind the Simulation
For A-level students wanting more mathematical depth, there are several possibilities.
Population growth and decline
Let P(n) represent the number of living cells in generation n.
We can investigate how P(n) changes with time.
Does it:
fall to zero?
eventually become constant?
fluctuate periodically?
continue growing?
We can plot P(n) against n for different starting configurations.
A blinker has constant population:
P(n) = 3
A glider also has constant population:
P(n) = 5
However, a complicated initial configuration can produce a much less predictable population graph.
The important point is that a constant population does not necessarily imply a constant pattern.
Probability
Suppose every cell is independently alive with probability p.
For an interior cell on an initially random grid, the probability that exactly three of its eight neighbours are alive is:
P(exactly 3 neighbours) = 56 × p³ × (1 - p)⁵
The number 56 comes from choosing which three of the eight neighbours are alive:
8! / (3! × 5!) = 56
For a new live cell to appear, the central cell must also be dead.
Therefore:
P(birth at a cell) = (1 - p) × 56 × p³ × (1 - p)⁵
This simplifies to:
P(birth at a cell) = 56 × p³ × (1 - p)⁶
This formula describes the probability of a birth in the first update of the randomly initialised grid.
It does not automatically apply to every subsequent generation because neighbouring cells become statistically dependent as the simulation evolves.
That distinction is a valuable extension into probability and mathematical modelling.
Determinism
The Game of Life is deterministic.
If we start from exactly the same configuration and use the same rules and boundaries, the same sequence of generations will occur.
Yet complicated patterns may be extremely difficult to predict without actually running the simulation.
This illustrates the distinction between a system being deterministic and its long-term behaviour being easy to predict.
Students can investigate the sensitivity of particular patterns to small changes in their starting arrangements.
It is tempting to call every complicated outcome chaos, but mathematical chaos has a more specific meaning. Conway's Game of Life is best introduced as a deterministic discrete dynamical system with complex emergent behaviour.
14. Ten Mathematical Challenges for Students
For anyone who enjoys mathematical puzzles, here are ten investigations.
Challenge 1 — Extinction
Find the smallest non-empty starting pattern that completely disappears after one generation.
Challenge 2 — Stability
Construct a stable arrangement containing exactly six living cells.
Challenge 3 — The oscillator
Find a repeating pattern other than the blinker.
Challenge 4 — The glider
Demonstrate that a glider returns to its original shape after four generations, displaced diagonally.
Challenge 5 — Population
Find a pattern whose population increases during its first generation.
Challenge 6 — Initial conditions
Start with a blinker and add one extra living cell. Investigate how the behaviour changes depending on the position of that cell.
Challenge 7 — Collisions
Create two gliders travelling towards one another. What happens when they meet?
Challenge 8 — Randomness
Generate a 30 × 30 grid with 20% of its cells initially alive. Repeat with 40% and 60%. Compare the population after 50 generations.
Challenge 9 — Changing the rules
Modify the survival rule so that cells survive with two, three or four neighbours. How does the resulting world differ?
Challenge 10 — Design a machine
Investigate whether a system of moving patterns could be arranged to transport or represent a simple item of information.
These challenges range from accessible GCSE investigations to questions suitable for enthusiastic A-level Mathematics, Further Mathematics and Computer Science students.
Some can be answered using squared paper.
Others become much more interesting with a computer.
The important part is not simply obtaining an answer but explaining why the patterns behave as they do.
15. Why I Would Teach This at Hemel Private Tuition
Having taught mathematics and science for many years, I have found that some of the most memorable lessons begin with a question that does not initially look like an examination question.
Conway's Game of Life is particularly appealing because it is accessible without being trivial.
A GCSE student can understand the rules in minutes.
Within the same lesson, that student can explore coordinate grids, probability, pattern recognition, logical reasoning and mathematical proof.
An A-level student can go much further into discrete mathematics, combinatorics and dynamical systems.
A Computer Science student can create a working simulation, optimise the algorithm and experiment with more sophisticated cellular automata.
What makes this such an attractive teaching topic is that students can make their own discoveries.
They do not simply follow a worked example.
They ask questions, make predictions, test their ideas and discover that the results are not always what they expected.
There is also room for failure.
A carefully designed pattern may disappear after one generation.
Another may behave in a completely unexpected way.
That is not wasted effort. It is part of the investigation.
Although Conway's Game of Life is not generally a prescribed topic in GCSE or A-level Mathematics, exploring it helps develop skills that are valuable throughout those courses.
These include logical reasoning, clear explanations, systematic investigation, mathematical modelling and the ability to move between different representations of a problem.
At Hemel Private Tuition, I believe there is considerable value in occasionally going beyond the examination syllabus.
Examination success matters, but so does helping students recognise that mathematics is a creative and continually developing subject.
Sometimes the most effective way to build mathematical confidence is to let students explore something that does not have an obvious answer printed at the back of a textbook.
16. Try the Game of Life Yourself
The best introduction is to experiment.
Begin with three cells in a row.
Then try a square block.
Next, construct a glider.
Finally, create a random population and watch what happens.
Try making predictions before running the simulation.
An excellent starting point is the ConwayLife community website, which provides information, examples and links to simulation tools.
Useful resources include:
ConwayLife: https://conwaylife.com/
LifeWiki: https://conwaylife.com/wiki/Conway%27s_Game_of_Life
Game of Life online simulator: https://playgameoflife.com/
For students who enjoy coding, the next challenge is to create a simulator rather than simply using one.
Conclusion — Complexity Does Not Always Need Complicated Rules
There is something wonderfully surprising about Conway's Game of Life.
Four simple rules applied to a collection of squares can produce stability, movement, repetition, interaction and structures capable of computation.
Nobody has to instruct the glider to travel.
Nobody tells the blinker to oscillate.
Nobody directs the glider gun to produce its next glider.
Those behaviours arise from the rules themselves.
That is one of the most important lessons mathematics can teach us.
Complex behaviour does not necessarily require a complex explanation at the level of individual components.
Sometimes extraordinary outcomes emerge from very simple foundations.
And perhaps the most exciting question is not whether Conway's mathematical cells are genuinely alive.
It is this:
If four simple rules can create such an astonishing mathematical world, what other remarkable things might we discover by experimenting with mathematics?
Beyond the Syllabus at Hemel Private Tuition
At Hemel Private Tuition in Hemel Hempstead, I teach GCSE Mathematics, A-level Mathematics, Further Mathematics and Computer Science, helping students develop both examination skills and a deeper understanding of their subjects.
My teaching combines worked examples, investigation, technology and practical demonstrations wherever these can help make difficult ideas clearer.
Lessons are available in person and online.
Mathematics is much more than passing examinations. It is learning to think, question and discover.
Philip M Russell — Hemel Private Tuition
