First, why magnetic rather than (say) a set of optical beams
where interrupting any one of them, rings your bell? Is the
goal a functioning game, or is it to explore this particular idea?
You might be able to get even simpler if the Nerf dart has any
decent momentum remaining, like a suspended screen that is
made to contact a back comductive plate and close the circuit,
but hangs short of that at rest.
Magnet to induce meaningful current in a large coil, will have
to move fairly fast. I suspect a Nerf dart is not very.
You would have to apply some gain and likely some noise
rejection (ambient 60Hz hum is likely bigger than your slow
flying magnet's input, because it will fill the coil area while
the magnet is a tiny fraction making a tiny blip).
If time is your enemy then simplicity is your friend.