For which numbers is it possible to arrange a collection of exactly that number of pennies in the plane so that each penny touches exactly three other pennies?
You can just give this question to the participants and let them work. If you like you may give them pennies to experiment with.
We don't want to give away the answer, but here are some facts:
1. This is not possible for any odd number. (See the balloon math lesson)
2. It is possible for any sufficiently large even number. Thus the most interesting cases are small even numbers.
This is similar to the chromatic number question.
This activity was included in the University of New England Julia Robinson Mathematics Festival.
There is a mentor guide in addition to the participant hand-out attached.