無中文摘要

Lights out is a portable electronic game launched in 1995. Now it is divided into sigma-game and sigma^+-game in many literatures.
In this thesis, the main purpose is to simplify the calculation of sigma-game and sigma^+-game. It was going to solve a set of equations to check whether the configuration is solvable and get its solution, but it must solve (mn)^2 variables. Lights-chasing is a method to decrease the numbers of variables to mn variables.
Since traditional light-chasing method is not efficient and convenient, we give a new interpretation for light-chasing by Fibonacci polynomial and find its property, and then calculate their period in sigma-game and sigma^+-game in m x n grids respectively.

英文文獻:
[1] Marlow Anderson, Todds Feil, Turning Lights Out with Linear Algebra, Mathematics
Magazine, Volume 70,No. 4. October, 1998, Page 300-303
[2] John Goldwasser, Xinmao Wang, Yaokun Wu, Does the lit-only restriction make any
difference for the -game and +-game, European Journal of Combination, Volume
30, Issue 4, May 2009, Pages 774-787
[3] Said Kouachi, Eigenvalues and eigenvectors of tridiagonal matrices, Electronic Journal
of Linear Algebra, Volume 15, April 2006, Page 115-133
[4] Thomas Koshy, Fibonacci and Lucas Numbers with Application
[5] Tse-Kan Lin, Calculating the Upper Bounds the Commutative Puzzles in Mathematics,
Master thesis, NTNU, Taiwan, 2015
[6] J.Scherphuis, The mathematics of lights out, https://www.jaapsch.net/puzzles/lomath.htm