Most Clicks Problem in Lights Out
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Consider a game played on a simple graph G = (V, E) where each vertex consists of a clickable light. Clicking any vertex v toggles the on/off state of v and its neighbors. Starting from an initial configuration of lights, one wins the game by finding a solution: a sequence of clicks that turns off all the lights. When G is a 5 × 5 grid, this game was commercially available from Tiger Electronics as Lights Out. Restricting ourselves to solvable initial configurations, we pose a natural question about this game, the Most Clicks Problem (MCP): How many clicks does a worst-case initial configuration on G require to solve? The answer to the MCP is already known for nullity 0 graphs: those on which every initial configuration is solvable. Generalizing a technique from Scherphuis, we give an upper bound to the MCP for all grids of size (6k - 1) × (6k - 1). We show the value given by this upper bound exactly solves the MCP for all nullity 2 grids of this size. We conjecture that all nullity 2 grids are of size (6k - 1) × (6k - 1), which would mean we solved the MCP for all nullity 2 grids.
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