Skeleton Tower Investigation

The formula 2(n (n-1)) + n is simplified to equal 2n² − n

For Tower No.6 = 6 x 5 = 30                30 x 2 = 60                 60 + 6 = 66

2 x 6² = 72                 72 – 6 = 66  

        5        4        3        2        1                 The formula can be simplified:

        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.

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 number of cubes in the centre stack.

Pattern

Secondary Differences:

1. 2 = 2
2. 3 = 3
3. 4 = 4
4. 5 = 5

Using the secondary differences for these quadratics, we know the nth term will be:

a/2 n² + …The secondary difference is a and also the same as the number of wings in the shape.

The second part of the equation was found out using the differences between the sequences (number of cubes) and a/2 n².

5 Wings: nth term for Difference = n x1.5

3 Wings: nth term for difference = n x 0.5

Formulas

nth Term and Proof

‘Arms’ of the tower unite to form a rectangle with dimensions n by (n-1). The numbers of ‘arms’ are determined by the number of wings in the tower. Two ‘arms’ are needed to form a rectangle. For Example, a tower with three wings will form 1.5 (3/2) rectangles and the centre stack will be added. This formula can be simplified:

1.5 (n (n – 1) + n         

1.5 (n² - n) + n

1.5n² - 1.5n + n

1.5n² - 0.5n  

A tower with three wings and a centre stack of six.

 

         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

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.