The Game of Life

Probably a fair portion of you have heard the name John Conway in conjunction with a game called Life. It’s amazing how the few simple rules can generate tons of interest, research, and analysis.

Playing Life:

This isn’t a game where you take turns to try to obtain an objective; think of it more as a simulator. You start with a grid of cells, a universe (really small for this example). Fill in the cells however you wish. Empty cells are dead, while the grey cells are alive:

A game of Life, turn 0

Each cell has eight neighbor cells surrounding it. The following rules determine whether a cell will be alive on the following turn:

1. A dead cell with 3 living neighbors gains life.
2. A live cell has 2 or 3 living neighbors stays alive.
3. A live cell with 0 or 1 living neighbors dies of loneliness.
4. A live cell with 4 or more living neighbors dies of overcrowding.

So taking our starting position above:

What will happen for turn 1?

The green cells are dead cells that will live next turn, and the red cells will be killed.

The next few turns you can see here:

It’s still far from certain what the fate of this universe will be. Sometimes, all the cells will eventually die. Othertimes, the universe will remain stable, or continually expand. There’s no good way to predict how it will end up for an arbitrary starting position.

Another complexity is how we treat the edges of the universe. You could have the universe “wrap around”, so the cells on the far left and far right edges (as well as top and bottom) would actually be neighbors.

I could hack up a basic implementation of Life pretty quickly, but it’s been done so many times, I’ll just link you Johan Bontes’ program. According to the website it’s an awesome program (plus it’s free).

Or, if you have time and graph paper, I suppose you could do it by hand.

September 15th, 2007 5 comments
Posted by Donnie Filed under Mathematics