The movements of ions along the z axis of a quadrupole ion cage are simulated for the cases of superimposed hexapole and octapole fields. In contrast to a widely-held opinion, superimposed even-multipole fields, like octapoles, prevent ion losses by resonances, and may even be used to increase daughter ion yields. Superimposed odd-multipole fields, like hexapoles, may, be applied to build ion cages with fast unidirectional ion ejection scan methods, making use of the non-linear resonance effects of such ion cages. The value of this study is limited because only movements along the z axis were considered.