A series of activities to build up intuition on the mathematics of
Invent shapes with different numbers of stable and unstable equilibrium points
Imagine holding a set of point charges at fixed positions in space - you can choose how many, the positions and the sizes of the different charges.
Experiment with different configurations of charges to determine whether you could arrange the charges so that when released simultaneously none of them starts to move.
Could you arrange the charges so that when released simultaneously only $1$ or $2$ of them start to move?
You might use your intuition to explore various possible configurations, but you will need to use Coulomb's law to show that the vector force on a particular particle is zero (see the Notes below) to prove your results.
Imagine that you are allowed to hold fixed one or more of the charges. What are the possibilities now?
Do your configurations seem stable to slight perturbations, in that they won't fly apart when subjected to a slight nudge? How might you create a stable configuration to confine a charged particle?