Activity 2-1: The Game of Life www.carom-maths.co.uk Activity 2-1: The Game of Life

It was invented by John Conway The Game of Life is a fascinating simulation of how a population (of maybe bacteria?) might grow if subject to a few simple rules. It was invented by John Conway in 1970 and was gradually refined by him and his team until it reached the form here. The game is ideally suited to the computer, which can calculate successive generations of the population for us with ease. So what are these simple rules?

The action takes place on a square grid that shows the current generation of the population. Each square is either black (alive) or white (dead.)

If a square is dead to start with, then it can become alive Note that each square has eight neighbours.  If a square is dead to start with, then it can become alive in the next generation if it has exactly three live neighbours. (This is called the ‘birth’ rule.) Otherwise , it remains dead.

If, on the other hand, a square is live to start with, then if the square has 0 or 1 live neighbours, it dies (of ‘loneliness’?)

If the live square has 2 or 3 live neighbours, it remains alive in the next generation.

If the live square has 4 or more live neighbours, it dies (through ‘overcrowding.’)

Task: take some squared paper, Before turning to the computer, it is a good idea to try these rules out for yourself to see how they work. Task: take some squared paper, and take the following as your starting population. Use the rules to find the next four generations.

Configuration 5 just stays as it is. You should get this: Configuration 5 just stays as it is.

found themselves asking several questions; Conway and his team found themselves asking several questions; Are there starting populations that get bigger and bigger? Can we find a starting shape that moves across the page? (Conway called such a shape a ‘glider’.) Can we find a starting population that generates gliders? (Such a configuration would be a ‘glider-gun’.)

Task: now visit the link below for a computer program that will do the hard work for us, and try some shapes of your own before trying the suggested shapes. http://www.ies.co.jp/math/java/misc/life_game/life_game.html So shape A oscillates (period 2) while shapes B and C head towards a stable shape (the 2x2 square.) Shape D arrives at an oscillating pattern, while Shape E is a glider. Shape F is periodic, period 3, and is called the pulsar.

breeders, puffers, space fillers, Some recurring patterns have been given names. The patterns on the left are famous enough to have been named appropriately: try them out. Other types of configuration have been called breeders, puffers, space fillers, and so on.

So is the glider gun possible? Try this! This proves that some configurations can produce arbitrarily large numbers of live cells, something that Conway initially doubted.

Try out some shapes of your own. A huge range of possible behaviours are possible: you could try the ones below. Space filler http://radicaleye.com/lifepage/patterns/max.html Period-88 oscillator http://radicaleye.com/lifepage/patterns/p1/osc88.html Stable reflector http://radicaleye.com/lifepage/patterns/p1/stillref.html

Carom is written by Jonny Griffiths, mail@jonny-griffiths.net With thanks to: John Conway and his team. Wikipedia for another very helpful article. To IES for their excellent collection of Java applets. Paul Callahan for his terrific collection of Life material. Carom is written by Jonny Griffiths, mail@jonny-griffiths.net