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Skeleton Tower Investigation

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Introduction

Aimimage00.png

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

image12.jpg

Tower 1 – 1 cube in centre

O in wings

image13.jpg

Tower 2 – 2 cubes in centre

1 on each wing

4 on wings

image14.jpg

Tower 3 – 3 cubes in centre

3 on each wing

12 on wings

image15.jpg

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).

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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

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Conclusion

image04.pngimage02.pngimage08.pngimage09.pngimage07.pngimage06.pngimage08.pngimage03.pngimage01.png

         5       4            3         2        1image11.pngimage05.png

                                                               2

                                                                 3

                                                                 4

                                                                 5image04.png

                                                                 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

...read more.

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