Skip to 0 minutes and 0 secondsThe Game of Life is played on an infinite board. Obviously, we’ll have to pretend that this 10 by 10 board is infinite… Each square on the board is called a cell. An empty cell is a dead cell. A dark cell is a live cell. This board has one live cell and 99 dead cells. Every cell on the board has eight neighbouring cells. In the Game of Life, patterns evolve on the board according to a set of rules. The starting point of the game is an initial pattern, like this one. The rules are applied simultaneously to the pattern, so that a new generation is born. A live cell is born in an empty cell that has three live neighbors.

Skip to 0 minutes and 42 secondsThe cells flashing blue have three live neighbors. These are the cells where a new live cell will be born. This is the new generation of the pattern. There are two cases where a live cell unfortunately dies. It can die from isolation. This happens when a live cell has one or no live neighbors. The cells flashing red cells will die. This is the new generation of the pattern. A live cell who has four or more live neighbors will die from overcrowding. The cells flashing red cells will die. This is the new generation of the pattern. Recapping. Live cells are born in an empty cell that has 3 live neighbours.

Skip to 1 minute and 38 secondsLive cells that have 1 or less, or, 4 or more live neighbours, die. All other live cells survive. Let’s see what happens when all the rules are applied at once. The blue cells will be born. The red cells will die. Looking at the whole picture… and this is the new generation. What happens to patterns in the Game of Life in the long run? Extinction is one possibility. A square is called ‘still Life’. Nothing ever changes. No cells are born and none die. This is a periodic oscillator, with a period of 2. And this is a glider that travels across the board. Life is fascinating! There is a whole gallery of Life forms waiting to be explored…

The rules of the Game of Life

This video explains all the rules needed to explain the Game of Life. Once you’ve watched the video you can play life on one of the online Life simulators. It will give you a feel of the game.

When John Conway invented the game, he chose the rules of survival, death and birth very carefully, after checking many other sets of rules. Most sets of rules led to too many deaths, or too many births. The rules that Conway chose in the end turned out to be the most interesting. In fact, mathematicians proved that the Game of Life is so ‘complete’, that it is what’s known as a Turing machine

Even though there are only a few, very simple rules, it is surprising to see how many complicated and interesting patterns evolve in Life. Oscillators, Still Life and patterns that become extinct are all common in Life. Are there patterns where the number of cells grow infinitely? Initially, Conway didn’t believe this could happen, so he offered a prize of $50 to anyone who could prove or disprove the existence of an “infinite” form. In November 1970 a group of researchers from MIT, led by Bill Gosper claimed the prize. They discovered a life form, called a “glider gun”, a pattern that”fires” a new glider every 30 generations. Since each new glider adds five (new) live cells to the original form, the population of living cells grows forever. Here is a glider gun:

Glider gun

Methuselah patterns stabilize only after many generations. Here is an example of a Methuselah that stabilizes after 386 generations, to become four still Lifes and eight oscillators.

Methuselah A Garden of Eden is a pattern that does not have a predecessor (sometimes called a “father pattern”), so it can only be the 0’th generation. Here is a ‘Garden of Eden’:

Garden of Eden

Only a few “Gardens of Eden” have been found to date. Why is it so difficult to search for Garden of Eden patterns? Conway’s rules are easy if we want to go forward from one generation to the next, but it is much more complicated to do the reverse – and go back in time… In other words, given a certain pattern, it is very difficult to find out what the previous generation was. One of the reasons for this is that many times a given pattern will have two or more predecessors. My own puzzle ‘Retrolife’, which you can read about in the next step, is based on the idea of “Gardens of Eden’.

Challenge

The following Game of Life pattern evolves at some time into a distinct number. What is the number?

Game of Life Challenge

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This video is from the free online course:

An Introduction to Recreational Math: Fun, Games and Puzzles

Weizmann Institute of Science