Random Colorings of a Lattice of Squares in the Plane

Square cells, tesselating the plane in a lattice arrangement, will be colored black or white by a random process. The coloring tries to imitate the appearance of cells with statistically independent colors, with black and white equally likely. Here only a relatively small initial set of cells is colored independently; the remaining colors are then determined by solving a linear recurrence equation. In this way one obtains colorings which, for some value of n, have independent colors in every set of n cells. The value of n, which depends on the recurrence equation used, can be deduced from divisibility properties of certain polynomials.