The still-Life density problem and its generalizations

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.Comment: 29 pages, including many figures drawn as LaTeX "pictures

New directions in enumerative chess problems

Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves in the solution is (usually) unique but the order is not, and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Almost all enumerative chess problems have been ``series-movers'', in which one side plays an uninterrupted series of moves, unanswered except possibly for one move by the opponent at the end. This can be convenient for setting up enumeration problems, but we show that other problem genres also lend themselves to composing enumerative problems. Some of the resulting enumerations cannot be shown (or have not yet been shown) in series-movers. This article is based on a presentation given at the banquet in honor of Richard Stanley's 60th birthday, and is dedicated to Stanley on this occasion.Comment: 14 pages, including many chess diagrams created with the Tutelaers font

New upper bounds on sphere packings I

We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur