For each positive integer n > 2 is there a "perfect" n-cubed n-cube? Roland Sprague found the first "perfect" squared square.
https://en.wikipedia.org/wiki/Squaring_the_square
For each positive integer n > 3, is there an analogous "perfect" hypercubing of the hypercube in dimension n?
https://en.wikipedia.org/wiki/Hypercube
 A: From Wikipedia:
"Unlike the case of squaring the square, a hard but solvable problem, cubing the cube is impossible. This can be shown by a relatively simple argument. Consider a hypothetical cubed cube. The bottom face of this cube is a squared square; lift off the rest of the cube, so you have a square region of the plane covered with a collection of cubes.
Consider the smallest cube in this collection, with side c (call it S). Since the smallest square of a squared square cannot be on its edge, its neighbours will all tower over it, meaning that there isn't space to put a cube of side larger than c on top of it. Since the construction is a cubed cube, you're not allowed to use a cube of side equal to c; so only smaller cubes may stand upon S. This means that the top face of S must be a squared square, and the argument continues by infinite descent. Thus it is not possible to dissect a cube into finitely many smaller cubes of different sizes.
Similarly, it is impossible to hypercube a hypercube, because each cell of the hypercube would need to be a cubed cube, and so on into the higher dimensions."
