  The distinguishing number of a graph $G$, denoted $D(G)$, is the minimum
number of colors needed to produce a coloring of the vertices of $G$ so that
every nontrivial isomorphism interchanges vertices of different colors. A list
assignment $L$ on a graph $G$ is a function that assigns each vertex of $G$ a
set of colors. An $L$-coloring of $G$ is a coloring in which each vertex is
colored with a color from $L(v)$. The list distinguishing number of $G$,
denoted $D_{\ell}(G)$ is the minimum $k$ such that every list assignment $L$
that assigns a list of size at least $k$ to every vertex permits a
distinguishing $L$-coloring. In this paper, we prove that when and $n$ is large
enough, the distinguishing and list-distinguishing numbers of $K_n\Box K_m$
agree for almost all $m>n$, and otherwise differ by at most one. As a part of
our proof, we give (to our knowledge) the first application of the
Combinatorial Nullstellensatz to the graph distinguishing problem and also
prove an inequality for the binomial distribution that may be of independent
interest.
