The forces that drive sea urchin primary invagination remain mysterious. To solve this mystery we have developed a set of finite element simulations that test five hypothesized mechanisms. Our models show that each of these mechanisms can generate an invagination; however, the mechanical properties of an epithelial sheet required for proper invagination are different for each mechanism. For example, we find that the gel swelling hypothesis of Lane et al. (Lane, M. C., Koehl, M. A. R., Wilt, F. and Keller, R. (1993) Development 117, 1049-1060) requires the embryo to possess a mechanically stiff apical extracellular matrix and highly deformable cells, whereas a hypothesis based on apical constriction of the epithelial cells requires a more compliant extracellular matrix. For each mechanism, we have mapped out a range of embryo designs that work. Additionally, the simulations predict specific cell shape changes accompanying each mechanism. This allows us to design experiments that can distinguish between different mechanisms, all of which can, in principle, drive primary invagination.