# The Painted Cube - Maths Investigations

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Introduction

The Painted Cube. D.K

A cube is painted on all its faces. It is then cut into 27 identical cubes. How many cubes have paint on.

(a) 3 faces.

(b) 2 faces.

(c) 1 face.

(d) 0 faces.

A similar cube is painted on all six faces it is then cut into 64 identical cubes. How many cubes have paint on. (a) 3 faces.

(b) 2 faces.

(c) 1 face.

(d) 0 faces.

A cube made of 27 smaller ones has a length, width and height of 3. I know this because 3*3*3= 27.

3*3*3

3 | Faces. | = | 8 |

2 | Faces. | = | 12 |

1 | Faces. | = | 6 |

0 | Faces. | = | 1 |

Total. | = | 27 |

Already just by looking at the first cube I have realised that all the cubes will have 8 small cubes with three faces covered, noticed this as all cubes no matter what size always have 8 corners.

A cube split into 64 identical cubes has lengths widths and heights of 4*4*4.

3 | Faces. | = | 8 |

2 | Faces. | = | 24 |

1 | Faces. | = | 24 |

0 | Faces. | = | 8 |

Total. | = | 64 |

My 4*4*4 cube has proven that all the faces with three sides covered will be 8.

Therefore the formulae for three faces is 8.

I will now try and find a formulae for a 1 side painted cube, to do this I will use a 5*5*5 cube.

Middle

If I take a length of small cubes, as long as they are not all made up from the outer layer they will look like this.

Because there are two small cubes on the ends of the strand it will make the formulae n-2 and because the 0 sided faces will make up an area I will have to cube it, making the formulae for a 0 sided cube (n-2)3.

Using a 6*6*6 cube and my formulae’s I will check to see if my work for cubes is correct.

3 | Faces. | = | 8 |

2 | Faces. | = | A |

1 | Faces. | = | B |

0 | Faces. | = | ## C |

Total. | = |

A) 12(n-2)

12(6-2) = 48

B) 6(n-2)2

6(6-2)2 = 96

C) (n-2)3

(6-2)3 = 64

I know that my formulae’s have worked because 6*6*6 equals 216 and so do all of my answers 8, A, B and C when added together.

## Cuboids.

Now that I have my formulas for cubes I will attempt cuboids. I

Conclusion

L*W*H gives me the whole large cube, so if I take away 2 from each of them then it takes away the outer layer for me.

The formulae for 0 faces covered in paint is (L-2)(W-2)(H-2).

The formulae for 2 faces covered in paint is 4(L+W+H-6). I found this out because a length, width and height added together gives me one of four combinations to get all the cubes covered in paint. I then take away the 6 because you take away 2 away from each of the lengths, widths and heights.

To make sure all of my formulae’s work I will test them on 3*3*5 cuboid.

3 | Faces. | = | 8 |

2 | Faces. | = | A |

1 | Faces. | = | B |

0 | Faces. | = | ## C |

Total. | = | 45 |

A) 4(L+W+H -6)

= 4(3+3+5 -6)

= 4(5)

= 20

B) 2(LH+LW+WH) –2(-4L–4W–4H +12)

= 2(15+9+15) –2(-12-12-20 +12)

= 2(39) –2(-32)

= 78 – 64

= 14

C) (L-2)(W-2)(H-2)

= (3-2)(3-2)(5-2)

= (1)(1)(3)

= 3

To check my answers I will add up the totals which I got using my formulae’s and check to see if they mach the total of 3*3*5= 45. 8 + 20 + 14 + 3 = 45

I know that all of my formulas are correct because the total of my answers I got from using my formulas match the answer I got from multiplying 3, 3 and 5 the lengths of my cuboid.

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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