  We introduce higher-dimensional cubical sliding puzzles that are inspired by
the classical 15 Puzzle from the 1880s. In our puzzles, on a $d$-dimensional
cube, a labeled token can be slid from one vertex to another if it is
topologically free to move on lower-dimensional faces. We analyze the
solvability of these puzzles by studying how the puzzle graph changes with the
number of labeled tokens vs empty vertices. We give characterizations of the
different regimes ranging from being completely stuck (and thus all puzzles
unsolvable) to having only one giant component where almost all puzzles can be
solved. For the Cube, the Tesseract, and the Penteract ($5$-dimensional cube)
we have implemented an algorithm to completely analyze their solvability and we
provide specific puzzles for which we know the minimum number of moves needed
to solve them.
