**Abstract:** |
The lights out game is a combinatorial game on a graph. Its goal is to go from one given state to another given state via some operations determined by the graph. We will first formulate variants of this game in mathematical language. Then, we will show its connection with classical groups, dynamics, line graphs, root systems, and demonstrate the strategy of a proof of the solution to this game. If time permits, we will present some new combinatorics discovered along this line of research: cactus skeleton, hypergraph cohomology, mixed graphs, and nucleus of graphs. |