  Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which
the tiles rotate as they slide around. In 1974, Wilson classified the groups of
solutions to sliding block puzzles. We generalize Wilson's result to puzzles
like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are
subgroups of the generalized symmetric groups. Aside from two exceptional
cases, we see that the group of solutions to such a puzzle is always either the
entire generalized symmetric group or one of two special subgroups of index
two.
