  A scramble on a connected multigraph is a collection of connected subgraphs
that generalizes the notion of a bramble. The maximum order of a scramble,
called the scramble number of a graph, was recently developed as a tool for
lower bounding divisorial gonality. We present results on the scramble of all
connected subgraphs with a fixed number of vertices, using these to calculate
scramble number and gonality both for large families of graphs, and for
specific examples like the $4$- and $5$-dimensional hypercube graphs. We also
study the computational complexity of the egg-cut number of a scramble.
