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#### Article Title

Turning the Lights Out in Three Dimensions

#### Abstract

Tiger electronics now has an entire Lights Out series. The original version, solved by means of Linear Algebra by Feil and Anderson in October 1998 is a five by five grid of lights. Pressing a button results in a change of parity of that button and a change in parity of the north, south, east, and west neighbors of that light (if such neighbors exist). The object of the game is to get all of the lights turned off. Later, Tiger released its next version of the mind puzzle, Lights Out Cube, a cube in which the sides are three by three grids of lights. The parity-changing rule still applies, except this time if a light lies on the border of a face, pressing it will change all of its neighbors, including those on adjacent faces. Thus, in Lights Out Cube, pressing any button will always result in the change of parity of five buttons, itself and its four neighbors. Again, the game presents the user with a configuration of lights, some off and some on, and the objective is to turn all the lights out. We will present a complete solution to Lights Out Cube in a style similar to that used by Feil and Anderson; however, a lack in certain mathematical conveniences of matrices present in the original Lights Out solution will complicate the process for the cube of lights.

#### Author Bio

Jake Tawney completed "Turning the Lights Out with Linear Algebra"under the direction of Dr. Todd Feil in the fall of 1997 at DenisonUniversity. One year later, Jake extended the result to the three dimensional version, Lights Out Cube. In preparation for the Rose-Hulman Undergraduate Mathematics Conference, "Turning the Lights Out in Three Dimensions was written and presented in the Spring of 1999, again under the direction of Dr. Feil.In May of 2000, Jake will graduate with a double major inMathematics and Computer Science from Denison University in Granville, Ohio. Jake has accepted a graduate offer at The Ohio State University and will begin his study in June of 2000. His mathematical interests are in Abstract Algebra, mainly Group Theory, and in Number Theory, both Classical and Algebraic. While at Ohio State, Jake hopes to earn a PhD in Mathematics and a Masters Degree in Computer Science.

Kyen Waldron worked on the paper with Suzy Reichel in the NSF funded Reserach Experience for Undergraduates program at Tulane University during the summer of 1999. Our faculty advisor was James Rogers. She will graduate from the University of Oregon in June 2000 with a bachelor of arts in mathematics and will pursue a masters degree in mathematics from the University of Oregon beginning in fall 2000. She also plan to obtain a teaching certificate and will probably go into mathematics education.

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