We theoretically investigate the energy spectra of vertically coupled quantum rings (VCQRs) with a three-dimensional (3D) model under applied magnetic fields. Two interesting configurations, the disk- and the conical-shaped (DI and CO) small VCQRs are explored. The electron and hole energies are significantly dominated by the interdistance d which plays a crucial role in the tunable states of structures. For electrons, the energy state is with a nonperiodical transition among the lowest electron. For holes, it does also demonstrate the same behavior. The energy band gap of VCQRs oscillates non periodically between the lowest electron and holes states as a function of external magnetic fields. In addition, the oscillation is controlled by not only the topology of VCQR and the inner (and base) radius but also the variation of d. This investigation is constructive in studying the magneto-optical phenomena of the nanoscale semiconductor artificial molecules, in particular for the laser device applications.