Geometry

# Painted Cubes

You can build up squares from smaller squares:

1 × 1 square | 2 × 2 square | 3 × 3 square |

In a similar way, you can build up cubes from smaller cubes:

1 × 1 × 1 cube^{[1]} |
2 × 2 × 2 cube^{[2]} |
3 × 3 × 3 cube^{[3]} |

### Think / Pair / Share

We call a 1 × 1 × 1 cube a **unit cube**.

- How many unit cubes are in a 2 × 2 × 2 cube?
- How many unit cubes are in a 3 × 3 × 3 cube?
- How many unit cubes are in a
*n*×*n*×*n*cube?

Explain your answers.

### Problem 10

Imagine you build a 3 × 3 × 3 cube from 27 small white unit cubes. Then you take your cube and dip it into a bucket of bright blue paint. After the cube dries, you take it apart, separating the small unit cubes.

- After you take the cube apart, some of the unit cubes are still all white (no blue paint). How many? How do you know you are right?
- After you take the cube apart, some of the unit cubes have blue paint on just one face. How many? How do you know you are right?
- After you take the cube apart, some of the unit cubes have blue paint on two faces. How many? How do you know you are right?
- After you take the cube apart, some of the unit cubes have blue paint on three faces. How many? How do you know you are right?
- After you take the cube apart, do any of the unit cubes have blue paint on more than three faces? How many? How do you know you are right?

### Problem 11

Generalize your work on Problem 10. What if you started with a 2 × 2 × 2 cube? Answer the same questions. What about a 4 × 4 × 4 cube? How about an *n* × *n* × *n* cube? Be sure to justify what you say.

- Image by Robert Webb's Stella software: http://www.software3d.com/Stella.php, via Wikimedia Commons. ↵
- Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons. ↵
- Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons. ↵