OK, dummy! You’re ready to uncover the marvelous secrets of the cube. You know I’ll be talking of moves, which are nothing more than lists of rotations, applied to any of the sides of the cube, rotations which can have a clockwise or a counter-clockwise direction. You already know the meaning of these terms, I just made sure we’re on the same page.
Before we consider the moves, it will help to point out how I came up with these moves. The basic idea of the solution to Rubik’s cube, as you will see in a future article, is to break down the problem into subproblems. One such division is the one I dubbed layer by layer. “Layer by layer” considers the cube as being formed of layers to be solved separately. As you look at the cube, having chosen a top side, you can make two horizontal cuts, forming three layers. These are the layers we will solve in order. Solve the top layer, then solve the middle layer, then solve the bottom layer and you’ve solved the whole cube. With a catch, of course. When you solve the second layer, you have to make sure not to “un-solve” the first layer. And, when you solve the third layer, you must be careful not to “un-solve” the first and second layers.
Here are the moves that I chose as representative in the context of the “layer by layer” approach. These are not the only possible moves, but hopefully a simple enough subset of them.
Move 1: R’ T’ H B’ H’ B T R. Bring the three rightmost cubies of the top side down to the front side, (via a one-time counter-clockwise rotation of the right side), move the middle and bottom cubies to the left side (though I illustrate this by moving the top side in the opposite direction), place the remaining cubie back on the top side (now note that when you rotated the top side, you moved the lone lateral cubie onto the right side, so to bring it back up, you must rotate the hidden side), move the bottom cubie from the left side to the hidden side (by a counter-clockwise rotation of the bottom side), bring the top cubie back on the lateral side, bring the middle and bottom cubies back under the top cubie, onto the front side, (the move actually brings the top and bottom cubies in alignment with the middle cubie) and place the three cubies back on the top layer. Note how, at the end of the operation, the entire top layer is unchanged.
Effect: Moves the bottom center cubie of the left side to the middle layer on the same side, to the left of the center cubie. Also, swaps the colors of the cubie. Affects other cubies on the bottom layer, but this is not an issue at this point. It does not affect any other cubies on the middle layer.
Move 2: R’ B’ R B R F’ R’ F. Move the three rightmost cubies of the top layer to the front side, remove the bottom one (by moving it to the left side), put the remaining two cubies back in place. Next, we concentrate on placing the wandering cubie back in, but not the way you took it out. Move the cubie to the front side, put it back in its final position (this has the effect of moving the other two cubies out of the top side and onto the hidden side, but we’ll fix this soon), move it laterally by rotating the front side, bring its two partners from the hidden side onto the top side, then bring the remaining cubies from the left side to the top side.
Effect: Swaps three of the four corner cubies of the bottom layer. It leaves the bottom right corner cubie of the front side unchanged. The other three cubies are moved as follows. Imagine looking at the bottom side of the cubie, with the front side at the top. The cubie that is left unchanged is now at the top right of the side you’re looking at. The top left cubie moves to the bottom left, the bottom left cubie moves to the bottom right, and the bottom right cubie moves to the top left. The move has other effects, which do not affect the top and middle layers.
Move 3: R’ B’ R B’ R’ B2 R B2. Move the three rightmost cubies of the top side to the front side, move the bottom one to the left side, put the remaining two cubies back in place. Now we alter our position such that we place the wandering cubie back slightly differently. Move the remaining cubie to the hidden side, bring the two rightmost cubies of the top side to the front side, then bring the wandering cubie back under them, and put the three back in place.
Effect: Flips three of the four corner cubies of the bottom layer. Look at the bottom side, with the front side on top. The move flips the top right, bottom right and bottom left cubies, leaving the top left cubie unchanged. The flip is counter-clockwise (in the direction that you would loosen a screw). The move affects other cubies, but it does not affect the top and middle layers, nor does it affect the corners in other ways.
Move 4: L’ R F’ L R’ B2 L’ R F’ L R’. Bring the central vertical belt of the top side to the front side (the first and the second rotation), move the bottom cubie of the belt on the left side, put the remaining belt back on the top side. Now prepare to put the cubie back in position. Move the cubie from the left side to the right side (the 180 degree of the bottom side), bring the belt back on the front side, then move the cubie back under its two siblings, to recreate the belt. Finally, move the belt back on the top side. A word of caution is due. Don’t lose track of which side is your front side. Although the narrative suggests you’re working with the bottom side when you move the bottom cubie of the belt to the top side, you are actually working with the front side now. It is the center cubie of each side that identifies which side you are working with. Just don’t get confused when you try to understand the move.
Effect: Swaps three of the four edge cubies of the bottom side. Looking at the bottom side, with the front side on top, the middle-left cubie moves as center-bottom, the center-bottom cubie moves as middle-right cubie, the middle-right cubie moves as middle-left cubie. In the process, cubie that was middle-left prior to the move and the cubie that was center-bottom prior to the move both become flipped.
Move 5: H2 B’ L R’ H2 L’ R B’ H2. This move is a flavor of Move 4. If we ignore the first two rotations of Move 4, we end up with a swapping of slightly different cubies. The initial 180 degree rotation of the hidden side of this move compensates that situation, and yields a slightly different result as a bonus.
Effect: Swaps the same three out of four edge cubies of the bottom side. It does not, however, flip the cubies in place.
The moves will soon show their power, as we uncover the Solution.