Test graduate students' ability to construct explicit families of Calabi-Yau threefolds with prescribed Hodge numbers by applying toric geometry techniques: select reflexive polytopes or toric diagrams, build associated singular toric varieties, identify and perform crepant resolutions, compute Hodge numbers combinatorially (Batyrev formulas), and present explicit defining equations or fan data. Assess correctness of constructions, proofs of crepantness, computations of H^{1,1} and H^{2,1}, deformation/Kähler moduli analysis, and discussion of mirror pairs where relevant.