Nanoparticles under a few nanometres in size have structures and material functions that differ from the bulk because of their distinct geometrical shapes and strong quantum confinement. These qualities could lead to unique device applications. Our mass spectral analysis of CdSe nanoparticles reveals that (CdSe)(33) and (CdSe)(34) are extremely stable: with a simple solution method, they grow in preference to any other chemical compositions to produce macroscopic quantities. First-principles calculations predict that these are puckered (CdSe)(28)-cages, with four- and six-membered rings based on the highly symmetric octahedral analogues of fullerenes, accommodating either (CdSe)(5) or (CdSe)(6) inside to form a three-dimensional network with essentially heteropolar sp(3)-bonding. This is in accordance with our X-ray and optical analyses. We have found similar mass spectra and atomic structures in CdS, CdTe, ZnS and ZnSe, demonstrating that mass-specified and macroscopically produced nanoparticles, which have been practically limited so far to elemental carbon(1), can now be extended to a vast variety of compound systems.