83 research outputs found
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
On some points-and-lines problems and configurations
We apply an old method for constructing points-and-lines configurations in
the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to
appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric
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
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