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Project Summary

Difficulty	7
Time required	Long (a couple of weeks)
Prerequisites	To do this project you should enjoy solving puzzles and thinking in three dimensions. This project requires starting with Rubik's Cube in the solved position, so you will need to know how to solve the puzzle in order to do this project.
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsored by a generous grant from Motorola

Abstract

Objective

The goal of this project is to figure out how to make patterns with Rubik's Cube.

Introduction

Rubik's cube is an interesting 3-dimensional puzzle that challenges your spatial imagination and memory. Usually, the goal is to arrange the cube so that each side is a solid color, as shown in Figure 1. However, in this project, you'll be trying to figure out ways of making geometrical patterns with Rubik's Cube. Before we get to the patterns, we'll introduce some notation for referring to the different sides, and show you how the cube is put together.

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 abbreviations 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). You can purchase the cube with many different color patterns. The color pattern we will be using has the following pairs of colors on opposite sides:

• white and yellow,
• red and orange,
• blue and green.

Figure 2. Diagram of the core of a Rubik's cube. The core enables each side, row, and column of the cube to rotate.

Before we present the patterns, we need to introduce some more terminology, so that we can easily refer to 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 three 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'.
3. When a side is rotated twice (direction does not matter in this case), the shorthand notation for the move is the letter + 2. For example, if you're supposed to rotate the front side twice, the shorthand would be F2.

A Simple Pattern: Checkerboard with Colors from Opposite Sides

To get you started, we'll show you how to create the first pattern. The first pattern has a checkerboard on each side, with colors from each of the two opposite sides. Starting with the cube in the solved position, rotate each of the six sides twice, in opposite pairs. The checkerboard pattern can be generated in 12 quarter-turn moves (or 6 moves if you count each half-turn of a face as single move). The Java applet below illustrates how the pattern is created.

Can you figure out how to solve the remaining patterns on your own?

Another Simple Pattern: Four Center Spots

This pattern can be generated from the solved cube in 12 quarter-turn moves (or 8 moves, if you count half-turns of a face as a single move).

Figure 3. Four center spots pattern.

Another Simple Pattern: Six Center Spots

This pattern can be generated from the solved cube in 8 moves.

Figure 4. Six center spots pattern.

Another Simple Pattern: Six T's

This pattern can be generated from the solved cube in 14 quarter-turn moves (or 9 moves if you count half-turns of a face as single move).

Figure 5. Six T's pattern.

A More Challenging Pattern: Cross Pattern

This pattern can be generated from the solved cube in 16 quarter-turn moves (or 11 moves if you count half-turns of a face as single move).

Figure 6. Cross pattern.

A Challenging Pattern: Cube in Cube

This pattern can be generated from the solved cube in 18 quarter-turn moves (or 15 moves if you count half-turns of a face as single move).

Figure 7. Cube in cube pattern.

A Challenging Pattern: Cube in Cube in Cube

This pattern can be generated from the solved cube in 20 quarter-turn moves (or 17 moves if you count half-turns of a face as single move).

Figure 8. Cube in cube in cube pattern.

A Challenging Pattern: Stripes

This pattern can be generated from the solved cube in 20 quarter-turn moves (or 17 moves if you count half-turns of a face as single move).

Figure 9. Stripes pattern.

A Challenging Pattern: Advanced Checkerboard

This pattern can be generated from the solved cube in 20 quarter-turn moves (or 16 moves if you count half-turns of a face as single move).

Figure 10. Advanced checkerboard pattern.

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?
• Does the order of the pairs of moves matter in creating the checkerboard pattern?

Bibliography

• Here are some ways to solve the cube, with step-by-step instructions. These webpages will teach you move sequences that produce specific rearrangements of the cube. They may help you to devise strategies for producing geometric patterns with Rubik's Cube:
  • Beust, C., 2003. "A Rubik's Cube Solution That Is Easy to Memorize," [accessed January 3, 2007] http://beust.com/rubik/.
  • Brown, R.A., 2004. "Rubik's Cube: The One-Minute Solution," [accessed January 3, 2007] http://www.brownsconnection.netfirms.com/rubik's_cube.htm,
  • 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.
• 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.

Experimental Procedure

1. Study the geometric Rubik's Cube patterns presented in the Introduction (Figures 3–10). Remember that each pattern uses the solved cube as the starting point. See how many of the patterns you can figure out how to make on your own.
2. Asking yourself the following questions may help you as you try to figure out how to generate the patterns:
   1. Which pieces have moved from the original configuration?
   2. Which pieces have stayed the same?
   3. For the pieces that have moved, did they move from an opposite side or an adjacent one?
3. How many moves does it take to create each pattern?

Variations

• For a more basic project using Rubik's Cube, see the Science Buddies project What's the Fastest Way to Solve Rubik's Cube?
• For an advanced Rubik's cube experiment, see the Science Buddies project Devising an Algorithm for Solving Rubik's Cube.
• Advanced. If you have skills in computer programming, learn how to use the Java applet to demonstrate how your algorithm works. 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.
• Design an experiment to find out if experienced cube solvers are faster at solving the puzzle from one of the pattern positions or from a random position.

• 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: 2009-04-29 22:00:00

Career Focus

If you like this project, you might enjoy exploring careers in Pure Mathematics.

	Mathematician Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering.			Statistician Statisticians use the power of math and probability theory to answer questions that affect the lives of millions of people. They tell educators which teaching method works best, tell policy-makers what levels of pesticides are acceptable in fresh fruit, tell doctors which treatment works best, tell builders which type of paint is the most durable. They are employed in virtually every type of industry imaginable, from engineering, manufacturing, and medicine to animal science, food production, transportation, and education. Everybody needs a statistician!
	Math Teacher Math teachers love mathematics and understand it well, but much more than that, they enjoy sharing their enthusiasm for the language of numbers with students. They use a variety of tools and techniques to help students grasp abstract concepts and show them that math describes the world around them. By helping students conquer fears and anxieties about math, teachers can open up many science and technology career possibilities for students. Teachers make a difference that lasts a lifetime!			Economist Every country has resources—people, land, raw materials, capital, and machinery—and economists study how those resources are distributed to create the goods that people buy, and the services people need or want. In their studies, economists monitor economic trends and collect data on things like energy costs, inflation, interest rates, exchange rates, business cycles, taxes, and employment levels. Based on their analysis of this data, they develop forecasts of economic activity so that businesses and governments can better plan for the future.

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