- Any live cell with two or three live neighbours survives.
- Any dead cell with three live neighbours becomes a live cell.
- All other live cells die in the next generation. Similarly, all other dead cells stay dead.
The Game of Life is undecidable, which means that given an initial pattern and a later pattern, no algorithm exists that can tell whether the later pattern is ever going to appear. This is a corollary of the halting problem: the problem of determining whether a given program will finish running or continue to run forever from an initial input.
The Algorithms section on the Wikipedia has some interesting snippets of information about the implementation. This is a neat way of calculating the next state of a cell;
if the sum of all nine fields in a given neighbourhood is three, the inner field state for the next generation will be life; if the all-field sum is four, the inner field retains its current state; and every other sum sets the inner field to death.