  Here we introduce a new game on graphs, called cup stacking, following a line
of what can be considered as $0$-, $1$-, or $2$-person games such as chip
firing, percolation, graph burning, zero forcing, cops and robbers, graph
pebbling, and graph pegging, among others. It can be more general, but the most
basic scenario begins with a single cup on each vertex of a graph. (This
simplification coincides with an earlier game devised by Gordon Hamilton.) For
a vertex with $k$ cups on it we can move all its cups to a vertex at distance
$k$ from it, provided the second vertex already has at least one cup on it. The
object is to stack all cups onto some pre-described target vertex. We say that
a graph is stackable if this can be accomplished for all possible target
vertices.
  In this paper we study cup stacking on many families of graphs, developing a
characterization of stackability in graphs and using it to prove the
stackability of complete graphs, paths, cycles, grids, the Petersen graph, many
Kneser graphs, some trees, cubes of dimension up to 20, "somewhat balanced"
complete $t$-partite graphs, and Hamiltonian diameter two graphs. Additionally
we use the Gallai-Edmonds Structure Theorem, the Edmonds Blossom Algorithm, and
the Hungarian algorithm to devise a polynomial algorithm to decide if a
diameter two graph is stackable.
  Our proof that cubes up to dimension 20 are stackable uses Kleitman's
Symmetric Chain Decomposition and the new result of Merino, M\"utze, and
Namrata that all generalized Johnson graphs (excluding the Petersen graph) are
Hamiltonian. We conjecture that all cubes and higher-dimensional grids are
stackable, and leave the reader with several open problems, questions, and
generalizations.
