  Given a finite abelian group $G$, consider the complete graph on the set of
all elements of $G$. Find a Hamiltonian cycle in this graph and for each pair
of consecutive vertices along the cycle compute their sum. What are the
smallest and the largest possible number of sums that can emerge in this way?
What is the expected number of sums if the cycle is chosen randomly? How the
answers change if an orientation is given to the cycle and differences (instead
of sums) are computed? We give complete solutions to some of these problems and
establish reasonably sharp estimates for the rest.
