The amyloid-β peptides form amyloid fibrils which are associated with Alzheimer's disease. Amyloid-β(29-42) is its C-terminal fragment and a critical determinant of the amyloid formation rate. This fragment forms the amyloid fibril by itself. However, the fragment conformation in the fibril has yet to be determined. The oligomerization process including the dimerization process is also still unknown. The dimerization process corresponds to an early process of the amyloidogenesis. In order to investigate the dimerization process and conformations, we applied the Hamiltonian replica-permutation method, which is a better alternative to the Hamiltonian replica-exchange method, to two amyloid-β(29-42) molecules in explicit water solvent. At the first step of the dimerization process, two amyloid-β(29-42) molecules came close to each other and had intermolecular side chain contacts. When two molecules had the intermolecular side chain contacts, the amyloid-β(29-42) tended to have intramolecular secondary structures, especially β-hairpin structures. The two molecules had intermolecular β-bridge structures by coming much closer at the second step of the dimerization process. Formation of these intermolecular β-bridge structures was induced by the β-hairpin structures. The intermolecular β-sheet structures elongated at the final step. Structures of the amyloid-β(29-42) in the monomer and dimer states are also shown with the free-energy landscapes, which were obtained by performing efficient sampling in the conformational space in our simulations.