Equivariant polynomial functions with the symmetries of the $n$-cube
are completely determined in terms of permutations of exponents.
Strategies for random searches of linear combinations of these
functions are described and used to generate interesting examples of
attractors. These attractors have symmetries that are an admissible
subgroup of the symmetries of the square, cube and 4-cube. A central
projection of the 4-cube with partial inversion is used for the
illustrations of attractors in four dimensions.