數學計算可交換性益智玩具的最少步數

Abstract

這篇論文的主旨是要描述點燈遊戲，及其他類可交換性益智玩具的最佳解。在1998年，Anderson與Feil用線性代數找出了點燈遊戲的解法。2009年，Goldwasser等人證明了點燈遊戲在只能按亮燈的限制下與一般遊戲的可解性並無不同。2014年，Schicho和Top進一步討論更多點燈遊戲的變形。這些結果都相當程度地仰賴電腦運算。在這篇論文中，我們試著用純數學方法去找出最佳解的上界，並給出上界的估計公式。

The main purpose of this paper is to describe the optimal solution of Lights Out games and other similar commutative puzzles. In 1998, Anderson and Feil used Linear Algebra to find a solution method for Lights Out games. In 2009, Goldwasser et al. proved the lit-only restriction is not different for the sigma game. In 2014, Schicho and Top discussed many variation of Lights Out. Those results heavily rely on computer. In this paper, we use mathematical methods to find an upper bound of minimal solutions, and furthermore, give an estimation algorithm to the upper bound.