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According to Van der Waerden, the sum of two natural numbers is defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) The product of two natural numbers is recursively defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) x^+ ( y^+ ) here means the successor(consequent) of x( y) in the set of natural numbers.

The paper mainly discusses, based on the properties and related knowledge of continuous functions, the conditions that the graphs of separated binary polynomials are closed curves by the method of separation of variables. It also studies the properties of closed curves in the ways of analytic geometry, and draws a sucient and necessary condition to determine whether the graph of a separated binary polynomial is closed.

Lights Out is a video game, which was released by Tiger Toys in 1995. Generally, there are 3×3, 4×4, 5×5, 10×10 grids of games. Under the normal circumstances, it’s easy to deal with some special configurations manually. However, the possibility of solving the randomly-generated configuration by hand is small, for it’s a bit difficult to find the law. Out of interest and enthusiasm about the game, our team reads some reference and gathered four mathematic methods to solve the game. What’s more, we extend the methods to m×n games, the cycling game, the lit-only game and N-color game. We independently propose a method to classify all configurations in the game for 5×5, which can also be extended to general cases. We also put forward the issues about N-order hexagon games, and provide some feasible solutions.

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