Theses, Dissertations, and Projects

2009

Honors Thesis

Department

Mathematics and Statistics

Abstract

This thesis is devoted to the investigation of the mathematics behind a popular puzzle known as the snake cube. The puzzle traditionally consists of a string of 27 small cubes that, when folded correctly, form a larger 3 x 3 x 3 cube. About six different versions are sold commercially, but many more are possible. Each cube that is not a beginning or ending cube is between two other cubes; either these two cubes are adjacent to directly opposite faces, or they are not. If they are, we say that the cube in question is a "straight", and if not we say that the cube is a "bend". All forms of the puzzle must correspond to at least one Hamilton path through the 3 x 3 x 3 grid graph. A large portion of the work and research in this paper was completed in Professor Atela's MTH 227 class in the Spring semester of 2008. The only previous work done on snake cubes has been concerning "bent" Hamilton paths and cycles, described by Rusky and Sawada in their paper "Bent Hamilton cycles in d-dimensional grid graphs". A bent path consists of entirely "bend" cubes, and when unfolded will have a zig-zag appearance. This paper describes for which d-dimensional grid graphs a Hamilton cycle or Hamilton path is possible.The Smith Math Department posesses a 5 x 5 x 5 cube which unfolds into a bent Hamilton path, designed and built by previous students of Professor Atela. Of particular interest were cubes that would unfold into various knot configurations. This twist on the traditional snake cube puzzle is, to my knowledge, an original one. A large number of \Hamilton Knots" were found during my work in MTH 227. The existance of these cycles through three-dimensional grid graphs suggests further questions. Of any knot configuration, we can ask, "What is the smallest cube that can contain such a knot? Can the knot span the entire cube? Section II deals with the material in previously published work, while Sections III and IV are composed entirely of original work.