Problems
A still life is a non-empty pattern (it starts with at least one alive cell) that never changes. Show that a pattern consisting of a \(2\times 2\) square of alive cells is a still life.
An oscillator is a pattern that returns to its original state after some number of evolutions. Its period is the number of evolutions it needs to return to its initial state. Show that a \(3\times 1\) piece of alive cells is an oscillator, and finds it period.
What is the smallest number of alive cells that a pattern needs to start with in order for the pattern to never die off?
A connected still life is a still life where you can get from any alive cell to any other alive cell by moving through neighbouring alive cells (remember that cells may touch at corners and still count as connected), Find a connected still life of exactly \(10\) alive cells.
Show that for any given even number greater than or equal to \(4\), there is a connected still life with that number of alive cells.
Prove that \(6\) is the maximum possible number of living neighbours a dead cell can have in a still life, and show that this maximum can actually occur.
A spaceship is a pattern that, after a fixed number of generations, looks exactly the same as before, but in a different place. For example, the glider is the pattern shown below: it looks the same every \(4\) generations, and each time it has moved one square diagonally. It turns out this is a speed limit in the game! Show that no pattern can move \(2\) or more squares diagonally after \(4\) generations.
A phoenix is a pattern with the interesting property that all of its alive cells die after each generation, yet the pattern as a whole lives indefinitely. Show that if a phoenix is contained in some rectangle at the start, it can never extend more than once cell past this rectangle (i.e: a phoenix can’t expand forever). Below is a picture of a phoenix with period \(2\):
Prove that there is no oscillator of period \(4\) (i.e: the whole pattern repeats every \(4\) generations) which has exactly one cell that also has period \(4\).