The Rubik Cube
A Rubik Cube has 6 faces. Each face has 9 colored tiles in a 3 by 3 arrangement. The Cube appears to be made up of 27 smaller cubes (8 corner cubes, 12 edge cubes, 6 more cubes--one in the center of each face, and 1 cube (which doesn't actually exist) in the center of the Rubik Cube). When you spin a face of the Cube, some tiles move from the adjacent faces to other adjacent faces. The entire layer, of course, moves along with the face so that corner and edge cubes also move, rotating around the center. Corners and edges never mix. That is, for any move and any combination of moves, corner cubes always replace other corners and edge cubes always replace other edges. Corner and edge cubes move separatly (but not independently, of course).
To facilitate discussion it is usually best to have the Cube's 6 faces oriented in a (temporarily) fixed position--one at the top (T), one at the bottom (Bo), and 4 vertical sides (L=left, F=front, Ba=back, R=right). If you spin the F face a quarter turn clockwise, then tiles from the T, R, Bo, and L faces move to the R, Bo, L, and T faces respectively. You can solve any mixed Cube by spinning individual faces 1 quarter turn, 2 quarter turns, or 3 quarter turns clockwise (or counter clockwise). The notation we will uaually use to indicate a spin will be the letter of the face followed by the number of quarter turns clockwise. Here are some examples of a spin on the front face (F):
|a quarter turn clockwise = F1
a half turn (either direction) = F2
a quarter turn counter-clockwise = F3
On the other hand, the center tiles--on the center cubes--only spin in place. They never move from one face to another. Therefore we can refer to a face by the color of its center tile. On my Cube the blue and green faces are opposite each other. If the top face is blue, the bottom face will be green. White and yellow are opposites. If the front face is white, then the back face is yellow. Red (right side) and orange (left side) will also be on opposite sides of the Cube. These pairs never change. The blue/green pair, for example, can be top/bottom, left/right, right/left, back/front, etc., but they will always be on opposite sides. These color pairs can be used to designate the 3 dimensions of the cube. In this orientation Blue-Green is the verticle dimension, Orange-Red is the left-right horizontal dimension, and White-Yellow the horizontal dimension of depth.
Each corner cube can be identified by the three colors of its 3 tiles. Likewise, each edge cube can be identified by the two colors of its 2 tiles. Those colored tiles also identify their home position--at the intersection of the 2 or 3 faces which have those colors (on their center tiles). When all of the corner and edge cubes are in their home positions and properly oriented, then the 9 tiles on each of the six faces will be of the same color (the color of the center tile). The Cube is then "solved".
There are many ways to solve a Rubic Cube. Any (non-trivial) solution involves several steps. The purpose of each step is to (1) place a set of (typically 4) cubes on a target face or an "equator" (located between two faces), (2) in their home position, and/or (3) with the 2 edge tiles or 3 corner tiles all properly oriented so that they match the color of the center tile of each face. The path, for example, of a typical solution for beginners is:
|The 4 edge cubes on the first face
Step 1: place and orient them
(which will, with the center tile, form a cross)
Step 2: position them
The 4 corner cubes on the first face
Step 3: place, position, and orient them
The 4 edge cubes around the equator
the layer just below the first face
Step 4: place, position, and orient them
The 4 edge cubes on the last face
have already been placed
Step 5: orient them
(which will, with the center tile, form a cross)
Step 6: position them
The 4 corner cubes on the last edge
have already been placed
Step 7: position them
Step 8: orient them
When you hold the Cube in front of you, below eye level, you can see the top face (T) and the front face (F). You can also easily see the right (R) and left (L) faces by rotating the Cube slightly. Since the bottom (Bo) and back (Ba) faces are more difficult to view, we will orient the Cube at each step so that the target area is on the top (T) or front (F) faces and the spins are mostly of the T, F, R, and L faces. Occasionally we will spin the bottom (Bo) face, but never the back (Ba) face. Now let's try this path to a soultion:
Step 1. Position and align 4 edges into a cross on the top face.
Decide which (color) face you will begin with, the blue face, for example. Orient the Cube so that this face is at the top. Then look for the 4 edge cubes with blue tiles. We will position the blue tiles around the blue center tile to form a blue cross. At the same time we will align the other tile on each edge cube with the center tile of the same color.
Step 1a. First, let's place those four yellow edge tiles on the top (yellow) face OR the bottom face (green on my Cube). Imagine the top face to be a conveyor beltüilike ë˝ô]Ä§Äiüjand the bottom face to be an upside down conveyor belt. Count the yellow tiles already on the top and bottom. Then look for the others on the four sides of the Cube.
If they are on a vertical edge, rotate the face that will place their yellow tile on the top face or on the bottom face. But do it after you have moved the conveyor belt so that no edge corners with yellow tiles get displaced from that face. When all the edge cubes on vertical edges have been moved up or down, check the horizontal edges and rotate the face a quarter turn to place those edge cubes on a vertical edge. Move them to the top or bottom face just as before.
Now, spin the top face so that the (vertical) side tiles of as many of the yellow edge cubes as possible line up with their own colors (center tile) on the side faces. Those that do NOT line up should be flipped (180 degree spin) to the bottom face.
Spin that face so that one of the non-target colored tiles lines up with its center tile. And flip those edges (180 degree spin) up to the target face. Repeat for each target edge on the bottom face.
Step 1c. Now, with the target face at the top, think of a kai-ten sush shop. The top face is the conveyor belt. The four edges are sushi-people. The (standing) edges in the middle layer are customers (counter people) sitting at the counter by the convery belt. The edges in the bottom layer are people in the street outside the sushi shop (street people).
We are trying to move all target edges to the conveyor belt (to be sushi people) and line them up with the adjoining face colors. Nothing significant changes when we turn the top face. The conveyor belt just goes around and around. Same with the bottom face the street people just go around and around on the street outside the sushi shop. BUT when we rotate one of the adjoining faces one of the counter people hops up on the conveyor belt, the corresponding sush person hops down on the other side, that counter person leaves the shop, and a street person enters and sits at the counter.
Therefore we want to maneuver any and all target edges (people with the target color) that are not yet "in final position" to a stand-by position (=at the counter). "In final position" means (1) on the conveyor belt (sushi people) (2) in the proper relative order (3) with the target color up. The sushi people only have to be in relative order, because the converyor belt is free to spin around until they line up with their own colors.
From the counter people can hop up on the conveyor belt in two different directions. One of those two directions will land them with the target color up. Turn the conveyor belt before each hop, so that the target edge (counter person) lands in the proper position on the conveyor belt.
If mistakes are made target edges may get displaced from their proper positions. Just keep checking and maneuver them back in place.
Step 2. Position and align the corners on the first face.
Step 3. Position and align the middle layer.
Step 4. Position the edges into a cross on the top face.
Some edges will proably NOT be aligned properly. Don't worry that is the next step.
Step 5. Align the edges by color with the center tiles of the lateral faces.
Step 6. Position corners on the top face.
Step 7. Align the corners by color with the center tiles of their faces.
Here's a special Cube for Christmas: