chalkdust

Life Gardner regularly received topics for the column directly from their inventors. His collaborators included Roger Penrose, Raymond Smullyan, Douglas Hofstadter, John Conway and many, many others. His closeness to researchers allowed him to write about ideas that the general public were previously unaware of and share newly researched ideas with the world. In 1970, for example, John Conway invented the Game of Life, oen simply referred to as Life. A few weeks later, Conway showed the game to Gardner, allowing him to write the first ever article about the now-popular game. In Life, cells on a square laice are either alive (black) or dead (white). The status of the cells in the next generation of the game is given by the following three rules: 1. Any live cell with one or no live neighbours dies of loneliness; 2. Any live cell with four or more live neighbours dies of overcrowding; 3. Any dead cell with exactly three live neighbours becomes alive. For example, here is a starting configuration and its next two generations:

The first three generations of a game of Life.

The collection of blocks on the right of this game is called a glider, as it will glide to the right and upwards as the generations advance. If we start Life with a single glider, then the glider will glide across the board forever, always covering five squares: this starting position will not lead to the sad ending where everything is dead. It is not obvious, however, whether there is a starting configuration that will lead the number of occupied squares to increase without bound. Originally, Conway and Gardner thought that this was impossible, but aer the article was published, a reader and mathematician called Bill Gosper discovered the glider gun: a starting arrangement in Life that fires a glider every 30 generations. As Gosper’s glider gun. each of these gliders will go on to live forever, this starting configuration results in the number of live cells perpetually increasing! This discovery allowed Conway to prove that any Turing machine can be built within Life: starting arrangements exist that can calculate the digits of pi, solve equations, or do any other calculation a computer is capable of (although very slowly)! 47

spring 2016

Chalkdust, Issue 03

Popular mathematics magazine from UCL