As part of this experiment you have a large board, this represents the cells of our automata, the labelling is not important for this experiment. If you don't have a grid you can just lay out the counters in a square lattice or draw on on paper. You've also got some binary way to indicate state, two sides of a counter or presence/absence of a counter. We'll talk about 'alive', 'green', 'good', 'present', '1' and the opposite notions of 'dead', 'red', 'bad', 'absent', '0'. I'll try and stick to the former but may fail. We use flip to denote a change between these states and often 'birth' to mean 'dead' to 'alive' and 'death' for the opposite transition. Obviously our grid should go on ad infintum but we don't have a big enough box for that. You may choose any size you wish, 10 by 10 is small but use able. You may want to simulate this experiment on a computer too.

So what is the Game of Life?

It's worth knowing some of the History. Initially people wanted to design something that simulated 'life', mathematicians defined this to mean it can 'reproduce' and also simulate a computer.

They looked at things called cellular automata, we have a grid (normally square) and in each put in a counter to indicate if the cell is alive or dead. We then run a series of rules to determine the change between turns.

While it's called a game there's not any players, you choose an initial configuration of cells and then let the game play for ever. We also don't have a boundary, you can make things as big as possible and they can drift arbitrarily far.

The game is by Conway who wanted to make something "interesting and unpredictable", you'll find stable patterns, drifting patterns and cycles. It also turns out that it satisfies the two properties for 'life'. His 4 design principles were

1. No explosive growth.

2. There are small patterns with chaotic, unpredictable outcomes.

3. There are 'self-replicators'.

4. The rules are simple.

So the rules are. We start by picking some cells to be alive. We then work in 'turns' or 'ticks' we apply all the rules simultaneously so all 'births' and 'deaths' happen at the same time. The neighbours of a cell are those in the 8 cells around it, this includes the diagonal neighbours but not the cell its self.

1. Any live cell with 0 or 1 live neighbours dies of under-population/loneliness.

2. Any live cell with 2 or 3 live neighbours lives on.

3. Any live cell with 4,5 or 6 live neighbours dies of over-population.

4. Any dead cell with exactly 3 live neighbours becomes alive by reproduction.

5. All other dead cells stay dead.

To do this you'll need to think of a snapshot and turn over based on that. If you want to explore big patterns your best looking at a computer.

Why not try and look for a stable pattern, these are rather boring patterns that don't do anything, so we'll need every cell to have 2 or 3 neighbours, see if you can find some. Thinking about making things relatively symmetric is a good idea, lots of small stable shapes have rotational and/or mirror symmetries, can you find one only with a single diagonal mirror symmetry? (loaf or boat work) You'll need to test the shapes under one iteration, you could maybe let them run an initial guess a few times to see if it stabilises while they understand the rules, don't try it too long though because of number 2 in the motivations, it'll probably also 'vanish'.

0000

0110

0110

0000

Square

000000

001100

010010

001101

000000

Beehive

000000

001100

010010

001010

000100

000000

Loaf

00000

01100

01010

00100

00000

Boat

00000

00100

01010

00100

00000

Tub

One cheeky one that also works is

000

000

000

Vannished

No let's try and look at some patterns that move, another interesting type of pattern is an oscillator, these cycle round a few shapes and then return. You might have found some trying the previous exercise. If not here are some you might try and find.

00000 -> 00000

00000 -> 00100

01110 -> 00100

00000 -> 00100

00000 -> 00000

Blinker

000000 -> 000000

000000 -> 000100

001110 -> 010010

011100 -> 010010

000000 -> 001000

000000 -> 000000

Toad

000000 -> 000000

011000 -> 011000

011000 -> 010000

000110 -> 000010

000110 -> 000110

000000 -> 000000

Blinker

On a computer you might also like to try the following, warning they get bigger than 10 by 10!

000000000000000

000111000111000

000000000000000

010000101000010

010000101000010

010000101000010

000111000111000

000000000000000

000111000111000

010000101000010

010000101000010

010000101000010

000000000000000

000111000111000

000000000000000

Pulsar (period 3)

00000

01110

01010

01110

01110

01110

01110

01010

01110

00000

Pentadecathlon (period 15)

Now we've seen these we might believe everything eventually becomes one of these stable or oscillating things, if it did we could sort of draw a box for each pattern and trap it. We'll talk about some patterns that need a big box later but in fact that's not the case. If you run a few computer trials you'll find sometimes you can 'fire' alive cells off the screen. You could try and make a shape like this on the grid, the simplest has 5 live cells and fits in a 4x4 square.

00000 -> 00000 -> 00000

00100 -> 00000 -> 00000

00010 -> 01010 -> 00010

01110 -> 00110 -> 01010

00000 -> 00100 -> 00110 etc.

Glider

0000000

0010010

0100000

0100010

0111100

0000000

Lightweight Space Ship (LWSS)

For computer users you may want to try and come up with some 'hard to kill' paterns, these take a while to settle into a combination of these types, a few you can try are bellow.

00000

00110

01100

00100

00000

R-pentomino

0000000000

0000000100

0110000000

0010001110

0000000000

Diehard

000000000

001000000

000010000

011001110

000000000

Acorn

These are called 'Methusulas' and Diehard takes 130 turns to fully disapear, which is thought to be maximal among 7 cell starts. Acorn takes 5206 turns and fires off 13 gliders.

The sheet also contains several large patterns like the "Gosper Glider Gun" and the "Block-laying switch engine" which periodically fires off gliders and blocks respectively. There are even breeders which leave behind trails of guns as they move along each gun firing out gliders.