Algebraic Structure of the Varikon Box

Abstract

The 15-Puzzle is a well studied permutation puzzle. This paper explores the group structure of a three-dimensional variant of the 15-Puzzle known as the Varikon Box, with the goal of providing a heuristic that would help a human solve it while minimizing the number of moves. First, we show by a parity argument which configurations of the puzzle are reachable. We define a generating set based on the three dimensions of movement, which generates a group that acts on the puzzle configurations, and we explore the structure of this group. Finally, we show a heuristic for solving the puzzle by writing an element of the symmetry group as a word in terms of a generating set, and we compute the shortest possible word for each puzzle configuration.

Gluing and cutting cube tiling codes in dimension six

“…, where all sets on the right side are non-empty. (If V = V i ,a 1 ∪V i ,a ′ 1 for every i ∈ [4], then V = U (Q ) and there is nothing to prove.) Since, by Corollary 2, for every j ∈ {2, ..., k }, the codes V i ,a j i c and V i ,a ′ j i c are strongly equivalent, we may pass by gluing and cutting from V to a cube tiling code V , where V = V i ,a 1 ∪ V i ,a ′ 1 (compare Subsection 2.2).…”

“…Let us note that the problem of passing from a cube tiling code V to a given cube tiling code W by gluing and cutting is very similar (taking into account a manner of doing a single operation) to the popular 15-puzzle game ( [23]). (Recently, a three dimensional version of this game, called Varikon cube was examined by d'Eon and Nehaniv in [4].) In our gluing and cutting game the object is to reach a given cube tiling code V ⊂ S d by gluing and cutting starting from some other cube tiling code W ⊂ S d .…”