Hyper-Dimensional Rubik's Cubes

MagicCube4D is a four-dimensional Rubik's cube. It is an exact analogy in four dimensions to the original plastic three dimensional puzzle. A five dimensional version is available here.  These are Rubik's cubes of the form 3^d, with the original popular puzzle being 3^3. We label the puzzles like this because they are a d-dimensional cube broken into 3^d smaller pieces or "cubies" of the same dimension. For example, the 3D cube has 3^3 or 27 total 3-dimensional cubies. Each of the d-dimensional cubies could be considered to have its faces covered by stickers of one smaller (d-1) dimension. But each cubie also only exposes a subset of its stickers to the "outside", meaning these are the stickers you could see if you lived and operated in d dimensions. We can use the number of exposed stickers as a classification of cubie types. For the 3D case, the 27 cubies are broken into 4 types, those that expose 0 stickers, 1 sticker ("centers"), 2 stickers ("edges"), or 3 stickers ("corners"). Each sticker on a given cubie has its own color, so we could also call these 1-colored, 2-colored, etc. pieces. More...