Following is a generalized notation system which assigns names to the pieces of every puzzle and most likely every future puzzle created by the members of this community in a straightforward and precise manner. It will be used in the derivations of the permutation counts found in Permutations_Index.
We shall denote by the term family a complete group of pieces that can occupy the same positions on a puzzle throughout all possible configurations. For example, on a 5^4 cube, there are two families of 3-colored pieces and three families of 2-colored pieces.
The basic idea is to classify families of pieces by the dimension of the section they are located in, rather than by the number of facelets they have. To make this more explicit, consider the 4-colored corner pieces of the 3^4 Rubik's cube. If the cube were replaced by an actual tesseract, a corner would be a point, or dimension zero. So, we will refer to a corner piece as a (0D) piece. 3-colored pieces will be called (1D) pieces, 2-colored pieces will be called (2D) pieces, and 1-colored pieces will be called (3D) pieces. This classification of a piece by the dimension of the section it resides in will be called the type of that piece. It is easy to see how this generalizes to any other puzzle. For MagicTile, consider the order-5 Klein's Quartic. The 3-colored pieces are (0D) pieces, the 2-colored pieces are (1D) pieces, and the 1-colored pieces are (2D) pieces, of which there are two families, not counting the fixed centers. Also, the term region will refer to a section that contains connected pieces of a specific type. For example, on a 5^4 cube, a (1D) region is a 3×1 section that holds three (1D) pieces.
So far, this only appears to be a relabeling of the pieces based on their location on the puzzle, which seems redundant due to the fact that the number of facelets on a piece is itself determined by the location of that piece on the cube. However, the key idea is this: For (nD) pieces, where n >= 3, we continue to break down the location of the piece until we have n <= 2. For example, consider a 5^4 cube. Its (3D) pieces on a single face form a 3×3×3 cube. The pieces on the corners of that cube will be called (3D)(0D) pieces, the edges, (3D)(1D) pieces, and the faces, (3D)(2D) pieces, while the term (3D) pieces will continue to denote the entire group of pieces. The center piece will be referred to as a (3D) center, the term center always being reserved for a piece that lies at the center of the region it is located in. It should also be noted that on n^4 cubes for example, when n >= 6, all of the the (3D) pieces other than the center are broken down into either (3D)(0D), (3D)(1D), or (3D)(2D) pieces regardless of the depth of each piece within the face. For example, take a 7^4 cube. The (3D) pieces in the inner 3×3×3 section will be broken down in the same manner as if they were the (3D) pieces in the 5^4 cube in the example above.
There is another observation to be made regarding the concept of regions. When considering regions for a type of piece that is subclassified (i.e. (3D)(1D) and (3D)(2D) regions for 4-dimensional cubes, for example), a group of pieces of such a type will be broken into separate regions according to the dimension of the subclassification. To make this completely clear, consider a 7^4 cube. A single face has one (3D) region, which is the complete 5×5×5 section of (3D) pieces on that face, as expected. Also, that face has twelve (3D)(2D) regions: Six of them are 3×3 sections of (3D)(2D) pieces (in the outer layer of the (3D) pieces), and the other six are 1×1 sections of (3D)(2D) pieces (in the inner layer of the (3D) pieces). We count these regions as separate, so that each region is a 2-dimensional group of pieces, even though each 3×3 region is connected to the piece of the corresponding 1×1 region. We can also see that there are twelve (3D)(1D) regions, six of which are 3×1 sections of (3D)(1D) pieces, and the remaining six 1×1 sections of (3D)(1D) pieces. It should now be clear what is meant by a (3D)(2D) and (3D)(1D) center on a cube of any size.
This notation system, or one which uses similar concepts, is essential when considering n^d cubes when d >= 5. For example, if d = 6, we can have (5D)(4D)(2D) pieces, (4D)(1D) pieces, or even (5D)(4D)(3D)(0D) pieces!
There is one more distinction to be made. We have already defined what we mean by a center piece. The rest are either wings or normals. Wings are pieces that only occur on a ..(2D) (e.g., (2D), (3D)(2D), or (4D)(2D), or (4D)(3D)(2D), etc.) region. Consider that region isolated from the rest of the puzzle, and can only rotate about its center in the plane it resides in. A piece is considered a wing if it belongs to a family (as defined under this restriction) to which there corresponds a separate family which are mirror images of each other. For example, on a square n×n ..(2D) region, wings are the pieces that do not lie on the main diagonals of the region or, when n is odd, additionally on the lines that divide the region into four equal square quadrants. Pieces that are neither wings nor centers will be called normals, except for ..(0D) pieces, which can simply be referred to as pieces without confusion. It is important to distinguish between wings and normals because the permutations and orientations of each always need to be calculated differently.
To conclude, here is a list of all possible types of pieces on a 4-dimensional cube:
(0D) pieces (corners)