  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.
