Conway's Game of Life

The whole of Life is two numbers. B3: a dead cell with exactly three live neighbours is born. S23: a live cell with two or three neighbours survives, and otherwise dies. There is no other rule, no randomness and no player — you set the first generation and the rest is forced. Everything the panel counts, and every pattern in the dropdown, is a consequence of those two numbers applied to every cell at once.

What each symbol means

n

the number of live cells among a cell’s eight neighbours — the four edges and the four corners. Every decision in the game is made from this one integer.

B3

the birth rule. Exactly three: two is not enough and four is too many, and that sharpness is why patterns have crisp edges instead of smudges.

S23

the survival rule. A live cell with fewer than two neighbours dies as if isolated, with more than three as if overcrowded.

Phase

the panel’s own verdict on what the pattern is doing — Evolving, Still life, Oscillating or Extinct. It is a comparison against earlier generations, not a prediction.

Where the formula comes from

1. For every cell on the board, count the live cells among its eight neighbours. Do this for the dead cells too — birth is decided the same way as survival, and a cell that is not there yet still has neighbours.
2. Apply B3. A dead cell with n = 3 becomes live. Load the block and check: each empty cell touching it sees at most two live cells, so nothing is ever born beside it. That is the whole reason the block sits still.
3. Apply S23. A live cell with n = 2 or n = 3 stays. Each of the block’s four cells touches exactly three others, so all four survive, and the block reproduces itself exactly.
4. Do all of it at once. Every cell is judged against the same snapshot, then the whole board is replaced. Update cell by cell instead and you get a different, wrong game — the neighbour you already changed would feed back into the cell beside it. Simultaneity is not an implementation detail; it is part of the rule.

How to read what you see

Watch the Phase row rather than the grid. Load the block and it turns to Still life after one generation; the blinker takes two before it reads Oscillating; the diehard runs for 130 generations and then reads Extinct. Now load the glider and step it as long as you like. It stays Evolving for ever — and it is the most orderly thing in the menu, repeating its exact shape every four generations. The classifier compares the whole board against earlier boards, and a glider’s board never repeats, because the shape has moved one square diagonally. Order that travels is invisible to a test for order that stays put.

Assumes

A finite board, and one you do not fully control: each pattern brings its own grid size and overrides yours. The page opens at 30×30; choose Acorn and it becomes 50×50, set 70 by hand and it holds, then choose Diehard and it drops to 40×40. Conway’s rule is defined on an unbounded plane, so a pattern that reaches the edge here is no longer playing quite the same game.

Breaks when

The Phase row can never tell you a pattern will survive. Load the diehard: seven cells, which grow to twenty-four, wander for over a hundred generations and are gone at generation 130 — the name is earned. Nothing on screen before generation 130 hinted at that, and nothing could have. Life is Turing-complete, so asking "will this pattern die out?" is the halting problem wearing a different hat, and no algorithm decides it. What the panel answers is the decidable question — has this board repeated yet? — and it is careful to say Evolving rather than anything stronger while it waits.