Forma, Vol. 13 (No. 1), pp. 13-21, 1998
Original Paper

How to Solve the Game "LIGHTS-OUT"
—An Application of Homomorphism in Group Theory—

Takaomi Ishihara

Literature Department, Seitoku University, 550 Iwase, Matsudo, Chiba 271-8555, Japan

(Received September 1, 1997; Accepted April 3, 1998)

Keywords: Group Theory, Homomorphism, LIGHTS-OUT, 4:1 Correspondence, Minimum Steps, Rectangular Bulb-Arrangements

Abstract. We have examined the puzzle game "LIGHTS-OUT" designed recently by A. Olti and G. Benedek, and found that this game can be understood by use of group theory. Using an elementary concept in group theory, a homomorphism or a homomorphic mapping, we have succeeded to reveal the underlying principles governing the puzzle. By arranging 25 bulbs in a 5 5 square, we found that there is a 4 to 1 homomorphic mapping between switch patterns and light patterns, so that only one fourth of the possible light patterns can be successfully darkened. We have also proposed a scheme to solve the game, i.e. just drive the bright bulbs down to the bottom line and then compare them with eight patterns. Owing to the 4:1 correspondence, there are four distinct ways to win the game, so that we can easily find the minimum steps needed to solve a certain light pattern. We have also studied rectangular bulb-arrangements and found that there are cyclic features in the order of the kernel of homomorphism.