Each entry is an element of Z2, the field of integers modulo 2. Anderson and Feil found that in order for a configuration to be solvable (deriving the null vector from the original configuration) it must be orthogonal to the two vectors N1 and N2 below (pictured as a 5×5 array but not to be confused with matrices).  formula_1  In addition, they found that N1 and N2 can be used to find three additional solutions to a solution and that these four solutions are the only four solutions (excluding redundant moves) to the starting given configuration. These four solutions are X, X + N1, X + N2, and X + N1 + N2 where X is a solution to the starting given configuration.. An introduction into this method was published by Robert Eisele. "Light chasing" is a method similar to Gaussian elimination which always solves the puzzle (if a solution exists), although with the possibility of many redundant steps. In this approach, rows are manipulated one at a time starting with the top row. All the lights are disabled in the row by toggling the adjacent lights in the row directly below. The same method is then used on the consecutive rows up to the last one. The last row is solved separately, depending on its active lights. Corresponding lights (see table below) in the top row are toggled and the initial algorithm is run again, resulting in a solution. Tables and strategies for other board sizes are generated by playing "Lights Out" with a blank board and observing the result of bringing a particular light from the top row down to the bottom row. Once a single solution is found, a solution with the minimum number of moves can be determined through elimination of redundant sets of button presses that have no cumulative effect. If the 5×5 puzzle is unsolvable under legal game creation, two leftmost lights on the bottom row will remain on when all other lights have been turned off. Existence of solutions has been proved for a wide variety of board configurations, such as hexagonal, while solutions to n-by-n boards for n≤200 have been explicitly constructed. There exists a solution for every N×N case. It is solvable on any undirected graph, where clicking on one vertex flips its value and its neighbours. More generally if the action matrix is symmetric then its diagonal is always solvable.