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We show that minimal models of nondegenerated hypersufaces defined by Laurentpolynomials with a $d$-dimensional Newton polytope $\Delta$ are Calabi-Yauvarieties $X$ if and only if the Fine interior of $\Delta$ consists of a singlelattice point. We give a combinatorial formula for computing the stringy Eulernumber of $X$. This formula allows to test mirror symmetry in cases when$\Delta$ is not a reflexive polytope. In particular we apply this formula topairs of lattice polytopes $(\Delta, \Delta^{\vee})$ that appear in theMavlyutov's generalization of the polar duality for reflexive polytopes. Someexamples of Mavlyutov's dual pairs $(\Delta, \Delta^{\vee})$ show that thestringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and$X^{\vee}$ may not satisfy the expected topological mirror symmetry test:$e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity ofan additional condition on Mavlyutov's pairs $(\Delta, \Delta^\vee)$.

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality