This paper details some history of the 15 Puzzle, several proofs from other mathematicians regarding the 15 Puzzle, and several original proofs of some variations. Advisors/Committee Members: White, Donald.

▼ Society often divides the worlds of mathematics and the arts into two disjoint entities, refusing to accept that any overlap could exist between such different realms. Our goal is to explore the intersection of the scientific and the beautiful via a self-contained journey through the life and tessellating works of the artist M.C. Escher. We invite the reader to dive headfirst into the underlying structure of Escher's systematic creations, focusing specifically on Wallpaper Groups and Hyperbolic Constructions - the two concepts that are subtly present in many of Escher's famous works. Advisors/Committee Members: Lewis, Mark.

▼ In this paper, we investigate extremal volumes of slices of the n-dimensional unit cube. If a cube is sliced by a central hyperplane, the maximal and minimal volumes of intersection are known, but the arguments are much more complex than one would expect to see for such a straightforward, geometrical query. Furthermore, if we dictate that the hyperplane must be a certain distance t from the center of the cube, then very little is known about the optimal volumes of intersection. This paper presents a brief history of this problem, and then gives a full solution for extremal one-dimensional slices and a partial solution for extremal hyperplane slices, when t is greater than ½√(n-1). Advisors/Committee Members: Zvavitch, Artem.