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Symmetrizing elements of the kernel to characterize the solvability of the σ+ Lights Out puzzle on various geometrical arrangements

The σ+ Lights Out puzzle is a game played on some geometrical grid wherein there is a bulb and a switch on each tile and the switches are wired so that flipping any switch changes the state of not only the bulb in its own tile but also of those in all adjacent tiles. A geometrical arrangement is said to be completely solvable if every initial configuration of `on' and `off' bulbs has a unique configuration of `switch flips' to turn all the bulbs in the initial configuration off. By symmetrizing elements of the kernel of the σ+ linear map, I present a method by which geometrical arrangements which have axes/planes of symmetry may be characterized as completely solvable or otherwise. The advantage of this method is that it can be used to reduce the complete solvability characterizations of more complicated arrangements to those of simpler arrangements, and therefore can be used to categorize these arrangements in a manner that is less cumbersome than previous polynomial approaches. I apply this method to reduce the characterizations of diamonds and surface grids on cylinders, capped-cylinders, tori, cones, capped-cones and spheres to the characterizations of rectangular grids. I also demonstrate how this method of symmetrizing elements of the kernel can be used to uncover many of the well-studied patterns regarding the characterization of m x n rectangular grids.