Problems

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 \(4\).

A pattern \(P\) is called a garden of Eden if there exists no pattern \(P'\) distinct to \(P\) such that \(P'\) evolves into \(P\) after one generation. Show that a garden of Eden exists. You do not need to provide an example of such a pattern.