The still-Life density problem and its generalizations

Abstract

A "still Life" is a subset S of the square lattice Z^2 fixed under the transition rule of Conway's Game of Life, i.e. a subset satisfying the following three conditions: 1. No element of Z^2-S has exactly three neighbors in S; 2. Every element of S has at least two neighbors in S; 3. Every element of S has at most three neighbors in S. Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points closest to x other than x itself. The "still-Life conjecture" is the assertion that a still Life cannot have density greater than 1/2 (a bound easily attained, for instance by {(x,y): x is even}). We prove this conjecture, showing that in fact condition 3 alone ensures that S has density at most 1/2. We then consider variations of the problem such as changing the number of allowed neighbors or the definition of neighborhoods; using a variety of methods we find some partial results and many new open problems and conjectures.