# Skeleton Tower Investigation

Extracts from this document...

Introduction

## Aim

A skeleton tower is made up of a stack of cubes with 4 triangular wings on each long face of the cube. Different towers have different numbers of total cubes and the aim of the investigation was to find an nth term and explain the reasons behind it.

## Towers

Tower 1 – 1 cube in centre

O in wings

Tower 2 – 2 cubes in centre

1 on each wing

4 on wings

Tower 3 – 3 cubes in centre

3 on each wing

12 on wings

Tower 4 – 4 cubes in centre

6 on each wing

24 on wings

Tower No. | No. in central stack | No. in each wing | Total no. in wings | Total no. of cubes |

1 | 1 | 0 | 0 | 1 |

2 | 2 | 1 | 4 | 6 |

3 | 3 | 3 | 12 | 15 |

4 | 4 | 6 | 24 | 28 |

6 | 6 | 15 | 60 | 66 |

12 | 12 | 66 | 264 | 276 |

## nth Term and Proof

For Tower No.6 I counted the cubes in one section (15 for a 6-high tower), multiplied that by 4 since there were four sections (60 for a 6-high tower), and then added the cubes in the middle stack (66 for a 6-high tower).

Middle

2 2 (n (n – 1) + n

3 2 (n² - n) + n

4 2n² - 2n + n

5 2n² - n

6

For Example, a tower with the centre column of 6 cubes can unite its two ‘arms’ to form a rectangle with dimensions 6 by 5 = n (n – 1) and as there are four triangular wings, there are four ‘arms’. The formula n (n – 1) is multiplied by 2 to get the number of cubes for the four arms 2 (n (n – 1). The center stack n is added and the formula is simplified to 2n² - n.

Tower No. | 2xn² | 2n² - n | Prediction | Answer |

6 | 72 | 66 | 66 | 66 |

12 | 288 | 276 | 276 | 276 |

## Extension

The task in the extension was to investigate the different numbers of wings on towers with differing centre stacks. The aim was to work out an nth term and explain the reasons behind it. Ultimately, the aim was to find out a formula for a tower with x wings and n

Conclusion

5 4 3 2 1

2

3

4

5

6

For 5:

2.5 (n (n – 1) + n

2.5 (n² - n) + n

2.5n² - 2.5n + n

2.5n² - 1.5n

Similar to when the tower with three wings was simplified, the tower with five wings raises a pattern. When the brackets are multiplied out and the centre stack is added, the value for the second part of the formula can be found out. The number of cubes in the centre stack is n and the number of wings is x:

(x/2 – 1) n

This part of the formula is found during the simplifying process.

So the overall formula is:

x/2 x n² - n (x/2 – 1)

x – Number of wings

n – Number of cubes in centre stack

## Prediction

Tower No. | Wings | Prediction | Check |

4 | 5 | 34 | 34 |

3 | 3 | 12 | 12 |

Prediction

5/2 x 4² - 4(5/2 – 1) = 34

3/2 x 3² - 3(3/2 – 1) = 12

Check

One face consists of 3+2+1 = 6 cubes. 6 x 5 = 30 cubes and the centre stack of 4 cubes is added = 34 cubes.

One face consists of 2+1 = 3 cubes. 3 x 3 = 9. The centre stack of 3 cubes is added = 12 cubes.

Ravi Ramesh 9.6 |

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

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month