The ability to control scattering directionality of nanoparticles is in high demand for many nanophotonic applications. One of the challenges is to design nanoparticles producing pure high-order multipole scattering (e.g., octopole, hexadecapole), whose contribution is usually negligible compared with strong low-order multipole scattering (i.e., dipole or quadrupole). Here we present an intuitive way to design such nanoparticles by introducing a void inside them. We show that both shell and ring nanostructures allow regimes with nearly pure high-order multipole scattering. Experimentally measured scattering diagrams from properly designed silicon rings at near-infrared wavelengths (∼800 nm) reproduce well scattering patterns of an electric octopole and magnetic hexadecapole. Our findings advance significantly inverse engineering of nanoparticles from given complex scattering characteristics, with possible applications in biosensing, optical metasurfaces, and quantum communications.