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Diagonal Difference

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Introduction

Mustafa Rafik 11.7API

Maths Coursework-Diagonal Difference

Introduction        

We are given an eight by eight grid.   Find the diagonal difference of different size grids (For e.g. 3 by 3, 4 by 4) within the eight by eight grid, by multiplying the opposite corners which results in two answers, we then deduct these two

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Middle

28

She notices that when you multiply the opposite corners the difference between the products is 32.

For example                10 times 28 = 280

                        12 times 26 = 312

The diagonal difference is 312 – 280 = 32

Aim

I have been asked to investigate the diagonal difference of a three by three grid inside an eight by eight grid.  I will then try to find a formula which relates to the diagonal difference of each square, I will then further this investigation by trying to find the diagonal difference of a nine by nine grid and a ten by ten grid and find the formula and see if it is the same. I will also do an extension by doing a rectangle instead of a square and then find the diagonal difference and the formula for this.

I am going to find the formula by finding the diagonal difference of all the sizes within the eight by eight grid, and then try to find any patterns, which would help me in finding the formula by drawing a grid.  This method would be good because it will show me the diagonal difference of any square of any size, because I would just have to insert the value of n in the formula to find out the diagonal difference of that square, therefore they will not have to write out all the squares and work out the diagonal difference.

Prediction

I predict that each time you increase the number of sides of the square within the eight by eight the diagonal difference will increase.

Method

As I progress through this investigation my aim is to find a formula or general rule which will suite or satisfy and grid size.  I will do this by looking at the diagonal difference and then trying to find a pattern or a rule.  I will do this by starting with the smallest grid size gradually moving on to the largest in a methodical order, and from the information, I will come up a general rule or formula. If I can, I will then convert this information into graphs and pie charts to show this information.

Results

I will first find the sequence of the diagonal difference in all of the different size grids.

Two by two grids

I will start by looking at the two by two grid.

1

2

9

10


2 times 9 = 18

10 times 1 = 10

Diagonal difference: 18-10 = 8

The diagonal difference is eight

I have found out that the diagonal difference for the two by two grid is eight, but I will try another two by two grid just to check this.

3

4

11

12

4 times 11 = 44

12 times 3 = 36

Diagonal difference: 44-36 = 8

The diagonal difference is eight again.

I again found that the diagonal difference is 8 so I know that the diagonal difference of two by two grids is 8, so I assume that if I did another square then I will get the answer of 8 because both grids have gave me an answer of 8, but just in case I will do a final two by two grid to prove that the diagonal difference is 8.

21

22

29

30

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Conclusion

Nth term

1

2

3

4

5

6

7

Diagonal difference

8

32

72

128

200

288

392

                                                        24          40        56         72       288        392

                                                              16        16        16        16        16

This tells me that because there is a second difference this is a quadratic equation.  And because it is a quadratic equation, I will halve what I have found as the first part of my equation, which is 16: -

        16 divided by 2 = 8

This tells me that eight is in my equation.

Therefore, eight is the first part of my equation.

I now have to find the n part in my equation to complete my equation; I will draw a graph to do this.

Nth term

1

2

3

4

5

6

7

Diagonal difference

8

32

72

128

200

288

392

To get 8 from n=1 in the Nth term table, I have to times 8 times to get the first number which is 8, so n must be, n1, but it does not work for32, so I will try n squared.

I will try it for the second number to see if it works,

        8 times n squared

        8 times 2 squared

        8 times 4 = 32

This formula works, so I have now found the whole formula

The formula is!        

        8 times n squared

I will now define to explain what my formula means.

The formula means (defining n)

N= the size of the grid inside the eight by eight grid

8= the eight by eight grid

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