Paolo

Objective The goal of this project is to compare different methods for solving Rubik's Cube. Which method provides the fastest solution? Introduction Rubik's cube is an interesting 3-dimensional puzzle that challenges your spatial imagination and memory. The goal is to arrange the cube so that each side is a solid color, as shown in Figure 1.

Figure 1. Diagram of a solved Rubik's cube. The six sides are named in pairs—up-down, front-back, and left-right. The up (U), front (F), and right (R) sides are visible. The remaining sides—left (L), back (B), and down (D)—are shown by the projected images. Figure 1 also shows the labels we will be using when referring to sides of the cube. The six sides are named in pairs—up-down, front-back, and left-right. To refer to a specific side, we'll use the one-letter abbrevations shown in Figure 1 (U, D, F, B, L, R). The cube is built in such a way that each side, row, and column can rotate (see Figure 2). With a few turns, the colors can be thoroughly mixed up. How can you get all of the squares back to their original positions? It's quite a puzzle to get the colors arranged properly again!

Figure 2. Diagram of the core of a Rubik's cube. The core enables each side, row, and column of the cube to rotate. In this project you'll learn a seven-step strategy for solving the cube. With this strategy, how quickly can you solve it? Next, you'll do background research (on your own, or online) and come up with an alternate method of solving the cube. Which method will work fastest for solving the cube? Before we present the strategy, we need to introduce some more terminology, so that we can distinguish individual pieces on the cube. Rubik's cube is made of three different types of pieces. We will refer to them as center, corner, and edge pieces. The puzzle has six center pieces, one in the middle of each face. Each center piece has only one visible face. There are eight corner pieces on the puzzle. Each corner piece has three visible faces. The remaining twelve pieces are edge pieces, occupying the middle position along each edge of the cube. Each edge piece has two visible faces. Center Piece

Corner Piece

Edge Piece

# in entire cube

6

8

12

visible faces

1

3

2

location

For each step in solving the cube, specific sequences of moves come in handy. In order to summarize the move sequences efficiently, we will use a shorthand notation common among cubers. The shorthand notation is easy to learn. There are just two rules you need to know. 1. When a side is rotated clockwise one quarter turn, the shorthand notation for the move is simply the letter of the side. For example, if you're supposed to rotate the right side one quarter turn clockwise, the shorthand would be R. 2. When a side is rotated counterclockwise one quarter turn, the shorthand notation for the move is the letter + an apostrophe ('). For example, if you're supposed to rotate the right side counterclockwise one quarter turn, the shorthand would be R'. Step #1: Solving the Top Cross Pick one of the six sides to start with. Your goal for this step is to solve the four edge pieces on that side. Each of the edge pieces needs to match both of its colored sides. You should be able to figure this step out on your own. If you find that one of the edges is in the right place, only the colors are reversed, the move sequence below will fix that problem.

Shorthand: R' U F' U' The Java applet above illustrates flipping a single edge piece. Note: if you see a gray box with a red "X" in the corner, you will need to update your Java Runtime Environment in order to run this applet. Go to http://www.java.com to get the latest version. Step #2: Solving the Top Corners Next, you need to solve the corners on the top side. To insert a corner piece into its correct place in the top layer, position the corner piece in the bottom layer, directly below the desired corner. The move sequence shown below will put the corner in place. You may find that when the corner piece first goes in, it is not oriented properly. Just repeat the same sequence (either twice more or four times more) and the corner will be oriented properly.

Shorthand: R' D' R D The Java applet above illustrates inserting a single corner piece into the top layer. Step #3: Solving the Middle Layer Edges Important: turn the cube over, so that the side you just finished solving is now on the bottom. The next step is to solve the middle layer edges. There are two sets of moves (below) that are used for this step. On the left, you'll see what to do to insert the edge piece from the top layer of the front side. On the right, you'll see what to do to insert the edge piece from the top layer on the right side.

Shorthand: U R U' R' U' F' U F Shorthand: U' F' U F U R U' R' The Java applets above illustrate the moves to insert edge pieces from the top layer into the middle layer. Step #4: Solving the Top Cross (Again) Now it's time to solve the top cross again. The tricky part is to solve it without undoing all the progress you've made so far. All three of the illustrations below use the same move sequence, F R U R' U' Fi. The goal is to get all of the top layer edge pieces the right color. You'll get them in the right positions in Step #5. Study the different starting positions below. You'll want to turn the top layer of the cube to match one of these

starting positions. Notice that the sequence may need to be repeated multiple times before you have all the edge pieces on top.

Shorthand: U' F' U F U R U' Shorthand: U' F' U F U R U' R' R' The Java applets above illustrate the moves to put the correct edge pieces into the top layer. Shorthand: F R U R' U' F'

Step #5: Solving the Top Edges In the previous step, you made a cross on the top layer. In this step, you'll get all of the top edge pieces into their proper positions. Turn the top layer until the front edge piece matches the front side (as shown below). If all of the other edge pieces are not in place, use this sequence to change their positions. With this sequence, only the front edge piece stays put, the other three rotate counterclockwise.

Shorthand: R U R' U R U U R' The Java applet above illustrates solving the top cross without disturbing the other layers. Step #6: Positioning the Top Corners In this step, you'll get all of the corner pieces in the top layer into their correct positions. One or more of them may be twisted the wrong way, but we'll take care of that in the last step. The only thing to worry about here is to make sure that each of the corner pieces is in the right place. Keeping the unsolved layer on top, rotate the cube until you find a corner piece that is in the right position. Put that corner piece right in front (where the front and right faces meet), as shown below. The move sequence shown below, U R U' L' U R' U' L, will rotate the remaining three corner pieces within the top layer. The front right corner stays put. If none of the corner pieces are in the correct position, then do this sequence once, and look for the corner that is in the right place and proceed as directed above.

Shorthand: U R U' L' U R' U' L The Java applet above illustrates positioning the top corners without disturbing the other layers. Step #7: Solving the Top Corners

This is it, the final step in solving the puzzle! This step is just a little bit tricky, so it is important to follow the directions carefully, or you'll lose all your hard work and have to start over. Keeping the unsolved layer on top, rotate the cube until you find a corner piece that needs to be flipped (i.e., top color does not match top layer). Position that corner right in front (where the front and right faces meet, as shown below). From now on, you need to keep the cube in this orientation. Remember the color of the front center piece, and make sure to keep that piece in the front position from now on. Do the move sequence R' D' R D either two or four times, until the corner piece is oriented correctly. The other layers may appear to be getting scrambled, but don't worry about it. If you follow the instructions, it will all come back right. Next, rotate the top layer clockwise, until you find another corner that needs to be flipped (i.e., top color does not match top layer). In the example below, the top layer needs to be turned twice to find such a corner (frames 9 and 10). Again, do the move sequence R' D' R D either two or four times, until the corner piece is oriented correctly. Repeat this procedure until all of the corners have been flipped. Once all the corners are oriented correctly, you'll see that you can solve the puzzle completely by simply turning the top layer.