x_j^{a_{ij}},$$ where $a_{ij} \in {\mathbb{N}}$, so that there exist positive integers $q_j$ and $d$ so that $\sum_{j} a_{ij} q_j=d$ for all $i$ ( i.e., $F_A$ is quasihomogeneous). The polynomial $F_A$ cuts out a hypersurface $X_A := Z(F_A) \subset W{\mathbb{P}}^n(q_0, \ldots, q_n)$ of dimension $N=n-1$. Further assume that this hypersurface is a quasi-smooth Calabi-Yau variety (the Calabi-Yau condition is equivalent to $\sum_i q_i = d$ and see Section 2 for details about the quasismooth condition). Greene and Plesser proposed a mirror to $X_A$ when the polynomial $F_A$ was Fermat [@GP]. Their proposed mirror for the hypersurface $X_A$ was a quotient of $X_A$ by all its phase symmetries of $X_A$ leaving the cohomology $H^{n,0}(X_A)$ invariant. The problem was that their proposal does not work well for the case when $X_A$ was not a Fermat hypersurface. Berglund and Hübsch proposed that the mirror of the hypersurface $X_A$ should relate to a hypersurface $X_{A^T}$ cut out by $$F_{A^T} =  \sum_{i=0}^n \prod_{j=0}^n x_j^{a_{ji}}.$$ The hypersurface $X_{A^T}$ sits inside a different weighted-projective 4-space, $W{\mathbb{P}}^n(r_0, \ldots, r_n)$. Berglund and Hübsch proposed that the mirror of $X_A$ should be a quotient of this new hypersurface $X_{A^T}$ by a suitable subgroup $P$ of the phase symmetries. In several examples, 