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Conway Game of Life 2026-10-13

Conway's Game of Life is a cellular automaton in which a simple local rule produces patterns that remain still, pulse, or travel across a square grid. Choose the starting pattern; after that there are no player decisions. The Life blinker and Life glider below make the surprise concrete: the rule mentions only births and survival, yet repeated applications create a clock and a moving shape.

The standard board is infinite, with cells at ordered pairs

(x, y) \in Z^{2}

of integers. At generation

t \in {0, 1, 2, \dots}

let

a_{t} (x, y)

1

for a live cell and

0

for a dead cell. Count all eight cells sharing an edge or a corner, excluding the center itself:

n_{t} (x, y) = \sum_{i, j \in {- 1, 0, 1} (i, j) \neq = (0, 0)} a_{t} (x + i, y + j) .

(1)

The local Boolean function defining the next generation is

a_{t + 1} (x, y) = ⎩ ⎨ ⎧ 1, 1, 0, n_{t} (x, y) = 3, a_{t} (x, y) = 1 and n_{t} (x, y) = 2, otherwise .

(2)

This is the rule B3/S23: a dead cell is born with exactly three live neighbors; a live cell survives with exactly two or three. A live cell with zero or one, or with four through eight, dies. A dead cell with two stays dead. All counts and center states come from generation

t

: this is a synchronous update, even if an algorithm computes its cells one at a time.^[ref]

In particular, the two births in the Life blinker do not help any other cell survive until the following generation. Writing each answer over its old value and then reading that new value for another count changes the rule. The simplest correct algorithm reads one array and writes the next generation into a second array; swap their roles after the entire step. The one-row in-place Life update achieves the same rule with less auxiliary storage by preserving the old values that later counts still need.

A finite implementation must specify its boundary convention. Holding all cells outside the rectangle permanently dead can destroy a Life glider when it reaches the edge. Wrapping neighbor coordinates from one edge to the opposite edge allows interaction with patterns on the other side. Neither convention is the infinite board. The figures display finite windows into an infinite board initially dead everywhere except the stated live cells; the windows do not impose walls.

Conway's Game of Life illustrates emergence in a precise, modest sense: the Life glider's motion is a collective pattern derived from local birth and death, without a motion command. It has deterministic evolution; visually surprising behavior introduces no random choices. Nor do the words "alive" and "dead" make it an experimentally validated model of biology. The beauty is that the same rule yields a stable four-cell square, the Life blinker, and the Life glider: three qualitatively different behaviors from a handful of cells.

For the stable square, take the live set

{(0, 0), (1, 0), (0, 1), (1, 1)}

. Each live cell has three live neighbors, and every dead cell has at most two. The next generation is identical: this configuration is a fixed point of the global update function. In contrast, the Life blinker returns after two generations, and the Life glider returns only after allowing a translation.

The patterns are useful as small, exact checks of an algorithm: the square checks survival without births, the Life blinker checks births and deaths in one simultaneous step, and the Life glider checks several steps and diagonal neighbors. More elaborate constructions can encode data and assemble Boolean circuits; Johnston and Greene develop constructions that simulate arbitrary computation in their chapter 9. This is a statement about suitably designed patterns with enough space and time, not a claim that every starting pattern computes something useful or that a fixed finite board has unlimited computer memory.^[ref]

Martin Gardner introduced Conway's Game of Life to his Scientific American readers in October 1970, reporting that Conway selected the rules after experimentation with patterns. The aim was sustained, varied behavior rather than immediate extinction or trivially demonstrable unlimited growth. These were mathematical explorations of chosen rules.^[ref]

Gardner's hands-on procedure is still a useful experiment: place black counters for the current live cells; mark doomed counters by stacking another black counter; mark births with white counters; finish checking before removing the doomed stacks and turning white counters black. Count only the original black occupants. Repeat with three adjacent counters to obtain the Life blinker, then with the five coordinates of the Life glider. Recording each generation tests whether the procedure implements the rule; it does not measure a natural population.^[ref]

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Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 5 / 10 / d / Solution Created 2026-09-24 Updated 2026-10-14

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A one-row in-place Life update needs only one saved row plus a few running sums. Before processing row

i

, above[j] stores the old row

i - 1

, and all rows

i

and below are still old. Let a column sum be the number of old live cells in that column in rows

i - 1, i, i + 1

. The three adjacent column sums, minus the center, give the required neighbour count.

java
static int columnSum(boolean[][] board, boolean[] above,
                     int i, int j) {
    int total = above[j] ? 1 : 0;
    if (board[i][j]) total++;
    if (i + 1 < board.length && board[i + 1][j]) total++;
    return total;
}

static void stepInPlace(boolean[][] board) {
    int width = board[0].length;
    boolean[] above = new boolean[width];
    for (int i = 0; i < board.length; i++) {
        int left = 0;
        int middle = columnSum(board, above, i, 0);
        for (int j = 0; j < width; j++) {
            int right = j + 1 < width
                ? columnSum(board, above, i, j + 1) : 0;
            boolean oldCenter = board[i][j];
            int neighbours = left + middle + right
                - (oldCenter ? 1 : 0);
            boolean newCenter = neighbours == 3
                || (oldCenter && neighbours == 2);
            above[j] = oldCenter;
            board[i][j] = newCenter;
            left = middle;
            middle = right;
        }
    }
}

When column

j

is processed, left and middle were computed before their cells were overwritten. right is computed now from untouched column

j + 1

, including its untouched above[j+1] entry. Thus all three sums describe old cells. Saving oldCenter into above[j] preserves the old current row for the next row's processing; no later computation in the current row rereads that changed entry, because its old contribution is already in the running sums. This proves the inner-loop invariant and then the row invariant by induction.

Initially above is all false, representing the dead exterior above the board. The missing left and right columns contribute zero; the bottom row omits any row below it. Hence all four boundaries follow the required convention. The update takes $O (H W)$ time and one $W$ -element auxiliary boolean vector, which is exactly a 1000-element vector for the requested board. It uses one board array throughout and does not silently keep a second board or two auxiliary rows.

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