# Making Patterns with Rubik's Cube

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 Areas of Science Pure Mathematics Difficulty Time Required Long (2-4 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

## Abstract

This project challenges you to figure out how to make geometric patterns with Rubik's Cube. Leaving your cube in one of these positions makes it much more tempting to pick it up and 'fix' it. Can you figure out how to make a checkerboard, or a cube-within-a-cube? Can you make only the center piece a different color from the rest? Can you figure out how to solve the cube from these positions?

## Objective

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

## Credits

Andrew Olson, Ph.D., Science Buddies

General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.

### MLA Style

Science Buddies Staff. "Making Patterns with Rubik's Cube." Science Buddies, 23 June 2020, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p024/pure-mathematics/making-patterns-with-rubiks-cube?class=AQVKQwtLTO7U2-4soRJY9iIUSmu4n5_pAWKyGUZtO5pO4tqercWnN3VNHOEJiZLee0LFZhWZgasaYL1rAIc9EiGd7HRo4oJIk5R6lQ-PjY0mGQ. Accessed 25 Oct. 2020.

### APA Style

Science Buddies Staff. (2020, June 23). Making Patterns with Rubik's Cube. Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p024/pure-mathematics/making-patterns-with-rubiks-cube?class=AQVKQwtLTO7U2-4soRJY9iIUSmu4n5_pAWKyGUZtO5pO4tqercWnN3VNHOEJiZLee0LFZhWZgasaYL1rAIc9EiGd7HRo4oJIk5R6lQ-PjY0mGQ

Last edit date: 2020-06-23

## 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.
An easy way to remember this pattern is 'plus yellow' because the second color of each pair can be made by adding yellow to the first color. As you'll see in some of the more advanced patterns, the color pairs matter. For example, making a checkerboard using color pairs from adjacent sides is more difficult than using color pairs from opposite sides.

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 started, we recommend looking into the simple checkerboard pattern. You can do this with the link to "Pretty Rubik's Cube patterns with algorithms" in the Bibliography. Scroll down that page to find the animation for the checkerboard pattern. Now, see if you can find out how to solve the remaining patterns listed on your own, without using online help.

### 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).

## Terms and Concepts

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?

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## Materials and Equipment

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

## If you like this project, you might enjoy exploring these related careers:

### Industrial Engineer

You've probably heard the expression "build a better mousetrap." Industrial engineers are the people who figure out how to do things better. They find ways that are smarter, faster, safer, and easier, so that companies become more efficient, productive, and profitable, and employees have work environments that are safer and more rewarding. You might think from their name that industrial engineers just work for big manufacturing companies, but they are employed in a wide range of industries, including the service, entertainment, shipping, and healthcare fields. For example, nobody likes to wait in a long line to get on a roller coaster ride, or to get admitted to the hospital. Industrial engineers tell companies how to shorten these processes. They try to make life and products better. Finding ways to do more with less is their motto. Read more

## 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.

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