Simple puzzles; unfortunately, I do not know how to formulate them in a whimsical fashion suitable for a dinner, they very much sound like math puzzles.
Take n labeled points $x_1, \dots, x_n$ in the plane. How do you construct a n-gon $a1, \dots, an$ such that for all i, $x_i$ is the midpoint of $[a_i, a_{i+1}]$ (with the convention $a_{n+1}=a_1$ of course). I was surprised to come across this problem in the puzzle pages of Le Monde. I think non-mathematicians would have a hard time with it.
For mathematicians who don't already know it, the Sylvester Gallai Theorem can offer stimulating after dinner discussions (or during those long proctoring sessions).
A napkin should be enough for this one (if even needed!). Consider a map f from the plane to the reals such that the sum of the values of f on the vertices of any square is zero. Find all such maps.
