# Conway’s Game of Life > Patterns from a local rule - Date: 1970 - Era: Modern age - Domain: [Electricity & Electronics](https://lastcodex.org/en/domains/electricity/) - Rebuild stage: I (Survival) - Difficulty: 1/5 - URL: https://lastcodex.org/en/inventions/conways-game-of-life/ John Conway’s Game of Life follows living and dead cells on a square grid. A single local rule, applied everywhere at once, produces patterns that remain still, repeat or travel. You can work through the generations with counters or explore them on a computer. ![Conway’s Game of Life · Patterns from a local rule](https://lastcodex.org/_astro/plate.BZzSyd0c_22S0tG.webp) ## Legend 1. Central cell and its eight neighbours 2. Initial horizontal row 3. Vertical row in the next generation 4. Stable block of four cells ## What it is The Game of Life is a cellular automaton devised by the mathematician John Horton Conway. It consists of cells whose states change according to a shared rule. Each square in the grid is either alive or dead; once an initial arrangement has been chosen, it determines the generations that follow. Birth and death are names for changes of state in this mathematical model inspired by the development of populations. There are no moves to choose while a pattern evolves. After setting it up, you apply the same rule at each generation. A board and counters let you do the calculations by hand. A computer makes larger patterns easier to explore, but the rule works the same way on either medium. ## How it works ![A framed slate on a wooden floor, chalked with three rows of three squares: a counter on the central square, the eight squares around it shaded blue, and a stick of chalk beside it](https://lastcodex.org/_astro/v1.C_wnGhPb_JiRGC.webp) _The diagonal squares count just as much as those sharing a side._ Every cell has eight neighbours: four sharing a side and four meeting it at a corner. The cell itself is excluded from the count. A dead cell becomes alive when it has exactly three living neighbours. A living cell survives with two or three living neighbours and dies in every other case. The notation B3/S23 records these conditions: B stands for birth with three neighbours, S for survival with two or three. Count all neighbours using the unchanged starting generation, then apply every change at once. A cell born during this update cannot contribute to another birth in the same update. Changing squares as you count would break this simultaneous evolution. The theoretical grid extends infinitely across the plane. On a finite board, you must decide what happens at the edges. In the examples below, the patterns have dead surroundings and opposite edges do not connect. The visible square runs from zero to six along each axis, with x increasing to the right and y downwards. Cells outside this window are dead, and the calculated patterns stay clear of its edges. ## How to do it To run the rule by hand, use a grid with counters or draw cells on squared paper. Keep the starting arrangement intact throughout the count and record the result on a second grid before moving to the next generation. This separate record lets you apply all births and deaths together, using the same starting state. Counters and paper are possible ways of representing the cells; the automaton is the rule connecting their arrangements. ![Three small slates on a low bench, linked by chalked arrows: three counters in a horizontal row, then in a vertical row, then in a horizontal row again](https://lastcodex.org/_astro/v2.7y7mQjfd_ZFhyV7.webp) _The same pattern returns after two generations without changing position._ Start with living cells at (2, 3), (3, 3) and (4, 3), leaving every other cell dead. Each end cell has only one living neighbour and dies. The middle cell has two and survives. The dead squares at (3, 2) and (3, 4) each have three living neighbours and become alive. The next generation therefore contains exactly (3, 2), (3, 3) and (3, 4). One more update restores exactly (2, 3), (3, 3) and (4, 3). This blinker has a period of two generations, with three living cells in each generation. Try the block separately, placing living cells at (2, 2), (3, 2), (2, 3) and (3, 3). Each of its four cells has three living neighbours, while no outside square has three. The block therefore remains unchanged in the next generation. An unchanging pattern is called a still life; one that returns cyclically is an oscillator. These two tests help check your diagonal counts and simultaneous updates before you explore other arrangements. ## History and evolution Cellular automata predate the Game of Life. Von Neumann was among the early researchers studying them, and these models were being explored in the early 1950s. The wider family includes different grids, states and neighbourhoods, beyond Conway’s particular rule. ![Five grids of six by six squares chalked on one long framed slate on a bench and linked by arrows, each holding five counters; the shape changes from grid to grid and on the last one it is back, one square lower and one square to the right](https://lastcodex.org/_astro/v3.CPhPFmWl_2mPhA9.webp) _Gardner's article already described patterns that move._ Conway arrived at his rules through lengthy experimentation. Martin Gardner introduced the Game of Life in the “Mathematical Games” column of the October 1970 issue of Scientific American. The article already described using counters, births and survival determined by neighbours, and patterns that stayed still, oscillated or moved. It also reported the use of a PDP-7 computer to investigate long sequences of generations, with a program by M. J. T. Guy and S. R. Bourne. Computer calculations subsequently extended the study of patterns and constructions. Related rules such as HighLife define other automata; Conway’s Game of Life retains its own rule. ## Key facts - Cell states: Alive or dead - Neighbourhood: 8 cells, including diagonals - Rule: B3/S23 - Publication: October 1970 ## Requires - [Computer](https://lastcodex.org/en/inventions/computer/): A machine with its program in memory (1948) - [Paper](https://lastcodex.org/en/inventions/paper/): A cheap surface for memory (c. 150 BCE) ## Sources - [The fantastic combinations of John Conway’s new solitaire game life](https://www.ibiblio.org/lifepatterns/october1970.html), Martin Gardner, Scientific American 223, octobre 1970, p. 120–123; transcription ibiblio - [Game of Life](https://mathworld.wolfram.com/GameofLife.html), Eric W. Weisstein, Wolfram MathWorld - [Mathematical Games — October 1970](https://www.scientificamerican.com/article/mathematical-games-1970-10/), Scientific American - [Cellular Automaton](https://mathworld.wolfram.com/CellularAutomaton.html), Eric W. Weisstein, Wolfram MathWorld Fact-checked on 2026-10-16. --- The Last Codex · https://lastcodex.org/en/inventions/conways-game-of-life/ · © The Last Codex. Texts, plates and figures: CC BY-NC-ND 4.0.