  The scramble number of a graph is an invariant recently developed to aid in
the study of divisorial gonality. In this paper we prove that scramble number
is NP-hard to compute, also providing a proof that computing gonality is
NP-hard even for simple graphs, as well as for metric graphs. We also provide
general lower bounds for the scramble number of a Cartesian product of graphs,
and apply these to compute gonality for many new families of product graphs.
