Abstract

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
location
# in entire cube	6	8	12
visible faces	1	3	2

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. Use the VCR-style buttons to navigate through the sequence.  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.

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.

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.

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.

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.

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.

Terms, Concepts, and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• Rubik's cube:
  • how it moves,
  • terminology:
    • corner pieces (8),
    • edge pieces (12),
    • center pieces (aka side pieces, 6).

Questions

• How many visible faces does an edge piece have? A center piece? A corner piece?

Bibliography

• Here are some other ways to solve the cube, with step-by-step instructions. You can find many more by searching online.
  • Beust, C., 2003. "A Rubik's Cube Solution That Is Easy to Memorize," [accessed January 3, 2007] http://beust.com/rubik/.
  • Lee, J., n.d. "Beginner Solution to the Rubik's Cube," [accessed April 29, 2009] http://peter.stillhq.com/jasmine/JasmineLeeBeginnerRubikSolution.pdf.
  • Petrus, L. 1997. "Solving Rubik's Cube for Speed," [accessed January 3, 2007] http://lar5.com/cube/index.html.
  • Youcandothecube.com. (n.d.). You CAN Do the Rubik's Cube. Retrieved November 23, 2010, from http://www.youcandothecube.com/.
• The Java applet used to illustrate the move sequences in this project is called "Caesar," and was written by Karl Hšrnell, Lars Petrus, and Matthew Smith. For instructions on using the applet, see the first link below. To download a copy of the applet, see the second link:
  • Petrus, L., date unknown. "The Java Cubes: How to Use the Cube Illustrations," [accessed January 3, 2007] http://lar5.com/cube/javacube.html.
  • Petrus, and Smith, 1996. "'Caesar' Rubik's Cube Applet (download page)," [accessed January 3, 2007] http://lar5.com/cube/downloads.html.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• a Rubik's cube,
• timer.

Experimental Procedure

1. Do your background research so that you are familiar with the terms, concepts, and questions, above.
2. If you can already solve Rubik's cube before doing this project, your first step should be to measure your average solution time. Time yourself for each of 5–10 trials, and calculate your average solution time. The cube should be well-randomized for each trial.
3. Study the strategy for solving the cube presented in the Introduction.
4. Practice using each set of moves on your Rubik's cube.
   1. Verify that each sequence works as described.
   2. Become familiar enough with each set of moves so that you can step through the sequence in your imagination, without having the cube in your hands.
   3. Practice solving the cube using the above moves.
5. Time yourself for 10 or more trials and see how long it takes you, on average, to solve the puzzle. The cube should be well randomized for each trial.
6. If you could solve the puzzle before you started the project, did your average solution time improve?
7. Repeat the process for a second method for solving Rubik's Cube.
   1. Study the method and verify that each sequence works as described.
   2. Become familiar enough with each set of moves so that you can step through the sequence in your imagination, without having the cube in your hands.
   3. Practice solving the cube. Allow yourself the same amount of time practicing as you used with the first method.
8. Time yourself for 10 or more trials and see how long it takes you, on average, to solve the puzzle with the new method. The cube should be well-randomized for each trial.
9. Which method is faster? Why?

Variations

• For a project on geometrical patterns with Rubik's Cube, see the Science Buddies project Making Patterns with Rubik's Cube.
• For an advanced Rubik's cube experiment, see the Science Buddies project Devising an Algorithm for Solving Rubik's Cube.

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas.

Credits

Andrew Olson, Ph.D., Science Buddies

Sources

The Java applet used to illustrate the moves in this project was written by Karl Hšrnell, Lars Petrus, and Matthew Smith. It can be obtained from: http://lar5.com/cube/downloads.html.

Last edit date: 2010-11-23 11:45:00