# Painted Cube Investigation

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Introduction

Investigation Rules

In this investigation I had to investigate the number of faces on a cube, which had had its outer surface painted red. I had to answer the question, 'How many of the small cubes will have no red faces, one red face, two red faces, and three faces?' From this, I hope to find a formula to work out the number of different faces on a cube sized 'n x n x n'.

The Rules and Solving the Problem

To solve this problem, I built different sized cubes (2 x 2 x 2, 3 x 3 x 3, 4 x 4 x 4) and made tables to help find the patterns.

Length of cube | No. of small cubes | No. of small cubes with painted faces | |||

3 | 2 | 1 | 0 | ||

2 | 8 | 8 | 0 | 0 | 0 |

3 | 27 | 8 | 12 | 6 | 1 |

4 | 64 | 8 | 24 | 24 | 8 |

No. of Painted Faces | 1 x 1 x 1 | 2 x 2 x 2 | 3 x 3 x 3 | 4 x 4 x 4 | n x n x n |

0 | 0 | 0 | 1 | 8 | (n – 2)3 |

1 | 0 | 0 | 6 | 24 | 6 (n – 2)2 |

2 | 0 | 0 | 12 | 24 | 12 (n – 2) |

3 | 0 | 8 | 8 | 8 | 8 unless 1 x 1 x 1 |

4 | 0 | 0 | 0 | 0 | 0 |

5 | 0 | 0 | 0 | 0 | 0 |

6 | 1 | 0 | 0 | 0 | 0 unless 1 x 1 x 1 |

Middle

No. of painted faces | Formula (n = 5) | Prediction | Answer |

0 | (n – 2)3 | 27 | 27 |

1 | 6 (n – 2)2 | 54 | 54 |

2 | 12 (n – 2) | 36 | 36 |

3 | 8 unless 1 x 1 x 1 | 8 | 8 |

4 | 0 | 0 | 0 |

5 | 0 | 0 | 0 |

6 | 0 unless 1 x 1 x 1 | 0 | 0 |

The table proves that the formulas are correct. All my formulae added together should be equal to the total number of small cubes in the large cube which is n3 . Here are the added formulae: n3 = (n – 2)3 + 6 (n – 2)2 + 12 (n – 2) +8

(n - 2)(n - 2)(n - 2) + 6(n - 2)(n - 2) + 12(n - 2) + 8

(n2 - 4n + 4)(n - 2) + 6(n2 - 4n + 4) + 12n - 24 + 8

n3 - 4n2+ 4n - 2n2 + 8n - 8 + 6n2 - 24n + 24 +12n - 24 + 8

n3 + 6n2 - 6n2 + 24n - 24n + 32 - 32

n3

Exceptions

None of the formulae will work for the cube 1 x 1 x 1: This pattern does not work because there is more than one vertex on one cube and therefore there is not going to be 8 cubes with 3 painted sides.

Conclusion

It is impossible for any cube to have 4 or 5 painted faces and for any cube bigger than a 1 x 1 x 1, it is impossible to have a cube with 6 painted faces.

Extending the Investigation

I extended the investigation to cuboids. I used measurements 2 x 3 x 4, 2 x 3 x 5 and 3 x 4 x 5.

Length of cuboids | No. of small cubes | Amount of small cubes with painted faces | |||

3 | 2 | 1 | 0 | ||

2 x 3 x 4 | 24 | 8 | 12 | 4 | 0 |

2 x 3 x 5 | 30 | 8 | 16 | 6 | 0 |

3 x 4 x 5 | 60 | 8 | 24 | 22 | 6 |

No. of Painted Faces | nth term | ||||

0 | (x - 2) x (y - 2) x (z - 2) | ||||

1 | 2 x ((x - 2)(y - 2)+(y - 2)(z -2)+(x - 2)(z - 2)) | ||||

2 | 4(x+y+z)- 24 | ||||

3 | 8 |

Exceptions to the rule are cuboids with one or two dimensions of 1.

Check

To check I used a 3 x 4 x 6 cuboid.

No. of Painted faces | Formula | Prediction | Answer |

0 | (x - 2) x (y - 2) x (z - 2) | 8 | 8 |

1 | 2 x ((x - 2)(y - 2)+(y - 2)(z -2)+(x - 2)(z - 2)) | 28 | 28 |

2 | 4(x+y+z)- 24 | 28 | 28 |

3 | 8 | 8 | 8 |

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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