Chapter 2-1: Cubies and Cubicles
Looking at the cube as a whole, at first glance, it appears to be made up of 3x3x3=27 cubies in three layers, each layer being a three-by-three square of small cubies. However, it is only possible to see the outside of the cube, so that only 26 cubies can be seen. The one in the center is only imaginary. Also, all that we can see of each of the 26 visible cubies are the colored facelets which combine to form the six faces of the entire cube. Each face of the cube is made up of nine such facelets. Thus there are 6x9=54 facelets on the cube.
| Terminology | Definition or Abbreviation |
|---|---|
| Cubies | The small cube pieces which make up the whole cube. |
| Cubicles | The spaces occupied by cubies. |
| Facelets | The faces of a cubie. |
| Types of Cubies: Corner, Edge, and Center |
A corner cubie has three facelets. An edge cubie has two facelets. A center cubie has one facelet. |
| Home Location -- of a cubie | The cubicle to which a cubie should be restored. |
| Home Position -- of a cubie | The orientation in the home location to which a cubie should be restored. |
| Positional Names for Cube Faces | Up/Down/Right/Left/Front/Back |
| Notation for Cubicles -- shown in italics | Lower case initials. For example, uf denotes the Up-Front edge cubicle. |
| Notation for Cubies -- shown in italics | Upper case initials. For example, URF denotes the cubie whose home position is in the Up-Right-Front corner. |
| Notation for Face Turns -- shown in BLOCK CAPITAL LETTERS | The initials, U, F, R, D, B, and L denote clockwise quarter turns. U-1, F-1, R-1, D-1, B-1, and L-1 denote counter-clockwise quarter turns. U2, F2, R2, D2, B2, and L2 denote half turns. |
| Moving the Whole Cube | 𝓤, 𝓕, 𝓡, 𝓓, 𝓑 and 𝓛 denote clockwise turns of the whole cube behind the indicated face. |
Look now at the cubies which make up the cube. Notice that the cube has three types of cubies. Some cubies have three visible facelets as indicated in Figure 2-2. These are called corner pieces. There are eight corners of the cube. Other cubies have only two visible facelets as indicated in Figure 2-3. These cubies fill in the space along an edge between two corner pieces. Therefore, they are called edge pieces. There are twelve edge pieces, one on each of the twelve edges of the cube. The third type of cubie has only one visible facelet. This facelet, as shown in Figure 2-4 is in middle of a face. Thus these cubies are called center pieces. There are six center pieces corresponding to the six faces of the cube.
By rotating different faces of the cube, the cubies can be moved about. Each cubie moves to the location vacated by another cubie. These locations are called cubcles. The locations occupied by corner cubies are corner cubicles and the locations occupies by edge cubies are edge cubicles. Observe that no matter how faces are rotated, the corner pieces always move from one corner cubicle to another corner cubicle and the edge pieces always move from one edge cubicle to another edge cubicle. Rotating a face never moves a center cubie from one face to another. The center pieces have a fixed location relative to the other center pieces. They can only be spun in place. This is a particularly important observation, because it shows the following:
The color of the center piece of any face defines the only color to which that face of the cube can be restored.
For each center piece the color of the opposite center piece never changes. Furthermore, if two opposite center pieces are placed in the positions of north and south poles respectively, then the sequential order of the other four center pieces around the equator is always the same.
Since the center cubie of each face determines the only color to which that face can be restored, we can also define the one and only cubicle in which each cubie can be placed to restore the cube. For example, if the two facelets of an edge cubie are orange and green, then that piece must be placed in the unique edge cubicle between the orange center piece and the green center piece as shown shaded in Figure 2-5. Furthermore, the cubie must be placed in that cubicle so that its orange facelet is next to the orange center piece and the green facelet is next to the green center piece.
Similarly, if the three facelets of a corner cubie are orange, green, and white then, to restore that cubie, it must be placed in the corner cubicle where the orange face, the green face, and the white face meet -- shaded in Figure 2-6. Furthermore, its orange, green, and white facelets must be on the orange, green and white faces respectively.
For each edge and corner cubie in the cube, the unique cubicle to which it must be restored is called the home location for that cubie. When a cubie is in its home lcoation and its facelets colors match the colors of the center pieces on each face, then the cubie is said to be in its home position.
It is possible for a cubie to be in the cubicle of its home location without being in its home position. A corner piece in this condition is said to be twisted in its home location. An edge piece in this condition is said to be flipped in its home location. Figure 2-7 shows a twisted corner cubie and a flipped edge cubie. Thus, each corner and edge cubie has a unique home location and in that cubicle it has a unique placement which puts it in its home position.
EXERCISES:
2.1-1 How many of the 54 facelets of the cube are a. facelets of curner cubies? b. facelets of edge cubies? c. facelets of center cubies?
2.1-2 At how many locations can an edge cubie be placed so that the colors of both of the two adjacent center cubies are different from both colors on the facelets of that edge cubie?
2.1-3 In what cubicle can a corner cubie be placed so that none of the center cubies adjacent to that cubiecles has the color of any of the three facelets of that corner cubie? Describe the cubicle location relative to the home position of the cubie.