Conway's game of life was invented by the mathematician John Conway in 1970. What's awesome about it? Touted as a zero player game with four basic rules this game has been used to prove several things including proving that the game is Turing complete (i.e. can complete any computation) and using it people have had insights into the origin of Life, The Universe and Everything
The game has a few really simple rules.
The Game of Life is played within a grid of squares. A few live cells are placed as initial configuration and as we go
The rules continue to be applied repeatedly to create further generations.
The cool thing is a simple configuration could produce runayway expansion or simply die off. And, according to the creator not even he can spot the difference. The patterns sometimes are complex and beautiful.
However the really cool part starts when a few mathematicians started producing logic gates using different confugurations of the game. They took it further and created a fully Turing Complete Machine. A life mimicking structure used to create a Turing Machine that can potentially handle any code. Eerily sounds like artificial life. The interesting part is a universe with such simple rules could be used to create complete computres.
Considering this is a simulation of basic life forms (super basic). Maybe Turing Machines can be a basis for us understanding how our brains work. That's still a long time coming and a large leap from this really awesome Math game.
Here are some resources about it