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 \Gamma 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 2 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 fx : 2jx1g). 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. 1. Life, the still-Life conjecture, Conjecture A, and ffi(n...

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