“关灯”游戏中的代数学

Algebra in the Game of ″lights-off″

研究的是来自互联网上的一个有趣的数学游戏,运用数学建模的方法给出了其基于有限域上的线性方程组的数学模型,对n=5的情形给出了所有解.并进一步,运用代数学的“分类”方法给出了游戏的四个等价类的直观描述,使游戏者不用解方程组就能立即判断出该“残局”能够变为哪个类型.

We study an interesting game from Internet, which is modeled as a linear equation system based on finite field F2= {0,1 }. We give all the solutions for case n=5. Furthermore, we describe the four equivalent classes of the game intuitively by the methods of Algebra, that makes the player determine easily which equivalent class the initial state of the chessboard belongs to, without solving the equation system.