  This article studies a generalization of magic squares to $k$-uniform
hypergraphs. In traditional magic squares the entries come from the natural
numbers. A magic labeling of the vertices in a graph or hypergraph has since
been generalized to allow for labels coming from any abelian group. We
demonstrate an algorithm for determining whether a given hypergraph has a magic
labeling over some abelian group. A slight adjustment of this algorithm also
allows one to determine whether a given hypergraph can be magically labeled
over $\mathbb{Z}$. As a demonstration, we use these algorithms to determine the
number of magic $n_3$-configurations for $n=7, \dots, 14$.
