Extended list of materials and tips:
- Four long strips of stiff, coloured paper (we found that paper with a grammage of 250 g/m2 and bands of length 40, 50, 60 and 70 cm worked best)
- A sheet of cardboard (or any other smooth, stiff material)
- Tape (strong enough to hold the rings steady as they wobble)
- Measuring tape
Delving a little deeper into the physics:
Resonance is getting the timing just right so that the energy you put into a physical system adds to the energy that's already there.  The most common example of resonance is a playground swing. A swing is characterized by its resonance frequency: if you push a person in a swing at a tempo that matches its resonance frequency, you make the swing go higher and higher, while attempts to push the swing at a faster or slower tempo don't work as well. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations. 
Resonance can only occur when a physical system is able to store and transfer energy between two or more forms of energy, for instance, in the case of the swing, between kinetic and gravitational potential energy. In the case of our paper rings, the transfer happens between kinetic and elastic potential energy: the rings are set in motion, which bends them, but having a certain stiffness, they tend to resist this bending to return to their equilibrium shape. Shaking a ring at a frequency close to its resonance frequency will boost the amplitude of the oscillation of that specific ring. As the other rings have different resonance frequencies, it will appear that this one ring is shaking while the others are still, similarly to how certain buildings collapse during an earthquake while others appear unaffected. [3,4]
Resonance is a phenomenon we put to good use in the superconducting radiofrequency cavities of the LHC.  They are hollow, ellipsoidal copper chambers coated with a thin film of superconducting niobium. Standing electromagnetic waves, oscillating at a set frequency of 400.8 MHz, are formed inside the cavity. The shape of the cavity is carefully designed so that at this specific frequency, each oscillation of the wave adds energy to the energy that is already there. In other words, they are designed to resonate at 400.8 MHz, which boosts the amplitude of the electromagnetic waves, making them larger and larger. This resonance is made possible because a transfer between electric energy and magnetic energy is constantly occurring inside the cavity, just like in an LC circuit studied in undergraduate classes. [6,7] This creates a strong electric field inside the cavity, and a strong magnetic field closer to its surface. While the magnetic field is not useful, the electric field is crucial to the LHC, as it is what accelerates the electrically charged particles injected in the particle accelerator.
Links for further information :
-  Steve Mould, A better description of resonance, YouTube (2017).
-  Wikipedia, Resonance.
-  Science Buddies & Megan Arnett, Ring on the Resonance!, Scientific American, Bring Science Home series (2016).
-  Exploratorium Science Snacks, Resonant Rings.
-  Home.cern, Accelerating: Radiofrequency cavities.
-  Wikipedia, Cavity resonator.
-  Wikipedia, LC circuit.