Computational understanding and manipulation of symmetries

For natural and artificial systems with some symmetry structure, computational understanding and manipulation can be achieved without learning by exploiting the algebraic structure. This algebraic coordinatization is based on a hierarchical (de)composition method. Here we describe this method and apply it to permutation puzzles. Coordinatization yields a structural understanding, not just solutions for the puzzles. In the case of the Rubik’s Cubes, different solving strategies correspond to different decompositions.