# Investigate different sized cubes, made up of single unit rods and justify formulae for the number of rods and joints in the cubes.

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Introduction

Aaron Evans 11SM Mathematics Coursework

Maths Coursework: Cubes, Rods, Cuboids

Introduction

I am going to investigate different sized cubes, made up of single unit rods and justify formulae for the number of rods and joints in the cubes. The cubes are made from single unit rods and are not hollow, meaning that the unit rods are constructed inside the cube making smaller, similar cubes inside of the default one. The only cube not to be made up of smaller cubes will be the 1x1x1 cube as this is the simplest form of cube and will, therefore not have any unit rods inside it. These cubes can be found on (sheet 1)

The individual unit rods in the structure are held together by a series of different types of joints, as shown below.

3 joints – found on the vertices of the cube and connect three different rods together.

4joints – found on the edges of the cube and connect four different rods together

5 joints – found on the faces of the cube and connect five different rods together

6 joints – found on the inside of the cube and connect six different rods together. Without using diagonals, this is the most amounts of rods to join together.

3 Joints | 4 Joints | 5 Joints | 6 Joints | Number of Joints | |

1 x 1 x 1 | 8 | 0 | 0 | 0 | 46 |

2 | 8 | 12 | 6 | 1 | 133 |

3 | 8 | 24 | 24 | 8 | 244 |

4 | 8 | 36 | 54 | 27 | 124 |

5 | 8 | 48 | 96 | 64 | 1046 |

The problem is to find formulae that represent the number of rods, 3 joints, 4 joints, 5 joints and 6 joints in an n x n x ncube. And then repeat for a cubiod

Cubes (Sheet 1)

I started the Investigation by drawing a cube shape.

Middle

Formulae

6 Joints = (n – 1) 3

Cubes: Rods

The next part of the investigation is to find out how many rods in each cube

1 x 1 x 1 Cube

This cube has 12 rods. I know this because each cube has six sides and the 1cm³ cube had 6 1cm square each square is made up of 4 rods, but some share a rod

6 x 2 = 12

2 x 2 x 2 Cube

This cube has 54 rods in total. I know this because there are 24 1cm square on the sides of the cube 24 x 2 = 48 rods the 6 other rods are inside connecting the sides together 24 x 2 + 6 = 54

3 x 3 x 3 Cube

This Cube houses 108 rods. Because there are 54 small squares with some that share rods on the cube 54 x 2 = 108

4 x 4 x 4 Cube

This cube has 300 rods. This is because there are 96 squares that make the faces of this cube with 2 rods each. 96 multiplied by 2 equal 192. Inside the cube there is another, which is the size equal to a 3 x 3 x 3 cube, which has 108 rods. 192 + 108 equals 300

5 x 5 x 5

This cube has 540 rods. This is because there are 60 rods on one face. 60 multiplied by the 6 faces that are on a cube. 60 x 6 equal 360. There are another 6 faces passing the opposite way with 30 exclusive rods. 30 multiplied by 6 equal 180. 360 +180 = 540.

Rods Formulae

From conducting the investigation I could see that the number of rods was equal to the number of 3 joints times 3 add the number of 4 joints times 4 add the number of 5 joints times 5 add the number of 6 joints times 6.

Conclusion

1 x 1 x 2 CUBOID

This cuboid has 20 rods. This is because it is made up of 2 1 x 1 x 1 cubes. These cubes are made up of 12 rods. 12 x 2 equals 24. As the cubes join in the centre there are four less rods. 24 minus 4 equals 20 rods.

2 x 2 x 4 CUBOID

This cuboid has 96 rods. This is because the two 2 x 2 x 2 cubes that it is made from have 54 rods each. Where they meet at the centre there are 12 less rods, as they makes up 2 faces. 108 minus 12 equals 96.

3 x 3 x 6 CUBOID

This cuboid has 264 rods. This is because the smaller face has 24 rods. There are 7 faces through the whole cuboid. 24 multiplied by 7 equals 168. Each larger face has 54 rods. As the vertical rods have been counted through the smaller face, only the horizontal rods count. There are 24 horizontal rods, and 4 larger faces through the cuboid. 24 multiplied by 4 equals 96. 96 plus 168 equals 264 rods.

4 x 4 x 8 CUBOID

This cuboid has 560 rods. This is because two 4 x 4 x 4 cubes that make up the cuboid have 300 rods each. The face has 40 rods. Where they meet in the centre the 40 rods make 2 faces, so 40 rods must be taken away. 600 minus 40 equals 560.

3 Joints | 4 Joints | 5 Joints | 6 Joints | Number of Joints | |

1 x 1 x 2 | 8 | 4 | 0 | 0 | 20 |

2 x 2 x 4 | 8 | 20 | 14 | 3 | 96 |

3 x 3 x 6 | 8 | 36 | 48 | 20 | 264 |

4 x 4 x 8 | 8 | 52 | 102 | 63 | 560 |

With this information I can calculate that a 100 x 100 x 200 cube will have 6,140,000 rods.

An n x n x n cube will have 6n3 + 10n2 + 4n rods.

Formula

Rods = 6n3 + 10n2 + 4n

To illustrate the increase of joints and the patterns that form I have made a graph

Also a graph to illustrate the increase in rods

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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