  Recall the classical 15-puzzle, consisting of 15 sliding blocks in a $4\times
4$ grid. Famously, the configuration space of this puzzle consists of two
connected components, corresponding to the odd and even permutations of the
symmetric group $S_{15}$. In 1974, Wilson generalised sliding block puzzles
beyond the $4\times 4$ grid to arbitrary graphs (considering $n-1$ sliding
blocks on a graph with $n$ vertices), and characterised the graphs for which
the corresponding configuration space is connected.
  In this work, we extend Wilson's characterisation to sliding block puzzles
with an arbitrary number of blocks (potentially leaving more than one empty
vertex). For any graph, we determine how many empty vertices are necessary to
connect the corresponding configuration space, and more generally we provide an
algorithm to determine whether any two configurations are connected. Our work
may also be interpreted within the framework of "Friends and Strangers graphs",
where empty vertices correspond to "social butterflies" and sliding blocks to
"asocial" people.
