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I am given a 10 by 10 grid. I am going to find the diagonal difference of different size grids (For e.g. 3 by 3, 4 by 4) within the 10 by 10 grid, by multiplying the opposite corners which results in two answers, we then deduct these two to get a final answer for that size.

This is the grid that I will use to help me investigate.

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Aim

I am going to investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 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 an 11 by 11 grid and a 12 by 12 grid and find the formula and see if it is the same.

I will also do an extension by doing a rectangle and 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 10 by 10 grids, 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 I will not have to write out all the squares and work out the diagonal difference.

Results

First I am going to try out all the squares then I will start with the rectangular.

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

I am trying to calculate the differences between these products.

I am going to multiply the opposite corners and then find the difference.

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12

21

22

11 x 22= 242

12 x 21= 252

The diagonal difference is 242- 252 = 10

I have found out that the diagonal difference for the two by two grids is 10, but I will try another two by two grid just to check this.

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16

25

26

15 x 26 = 390

16 x 25 = 400

Diagonal difference: 400 – 390 = 10

The diagonal difference is 10 again.

I again found that the diagonal difference is 10 so I know that the diagonal difference of two by two grids is 10, so I assume that if I did another square then I will get the answer of 10 because both grids have gave me an answer of 10, but just in case I will do a final two by two grid to prove that the diagonal difference is 10.

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37

38

27 x 38 = 1026

28 x 37= 1036

Diagonal difference: 1026- 1036 = 10

After doing this, I found out that the diagonal difference of two by two grids was 10 because all the two by two grids gave me an answer of 10.

Now I am going to try a 3 by 3 square grid.

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11 x 33 = 363

13 x 31 = 403

Diagonal difference: 40

Now I am going to try another 3 by 3 grid to show if my diagonal difference is correct.

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42 x 64 = 2688

44 x 62 = 2728

Diagonal difference: 40

The diagonal difference for a 3 by 3 grid is 40.

Now I am going to try out a 4 by 4 grid. My prediction is that the diagonal difference will be 90 because it is one below and then times by 10.

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55 x 88 = 4840

58 x 85 = 4930

Diagonal difference: 90

My prediction was correct. Now I am going to try another to make sure that I didn’t do anything wrong.

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11 x 44 = 484

14 x 41 = 574

Diagonal difference: 90

Now I am going to try a 5 by 5 grid.

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51 x 95 = 4845

55 x 91 = 5005

Diagonal difference: 160

Now I am going to try out work out the algebraic formula for working out the diagonal differences for all squares.

So for a 6 by 6 grid I predict that the diagonal difference would be; 250.

To show this I will do a number grid and also in algebra.

The general difference formula that I predict is (n-1)2 x 10.

Now to show that it works!

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45 x 100 = 4500

50 x 95 = 4750

Diagonal difference: 250

Now to show this in an algebraic form.

x

x+1

x+2

x+3

x+4

x+5

x+10

x+11

x+12

x+13

x+14

x+15

x+20

x+21

x+22

x+23

x+24

x+25

x+30

x+31

x+32

x+33

x+34

x+35

x+40

x+41

x+42

x+43

x+44

x+45

x+50

x+51

x+52

x+53

x+54

x+55

(x+50) (x+5)-x(x+55)

= x2+50x+5x+250-(x2+55x)

= x2+55x+250-x2-55x

= 250

My formula works. So if I were to do a 7 by 7 I would do 62×10.

Now I am going to put my results in a table.

Size of square

Differences

2 x 2

10

12×10

3 x 3

40

22×10

4 x 4

90

32×10

5 x 5

160

42×10

6 x 6

250

52×10

I have shown that for a 5 by 5 square grid you will have to 42×10 to get the answer, so if I wanted to do a 7 by 7 grid I would have to do; 62×10

7 x 7 it would be (7-1)2×10

= 62 x 10

=36 x 10

=360

As I have said before that the general formula is:

(n-1)2×10

So the grid would look like this:

Now I a going to try out the rectangular grids. For this I will try to use an algebraic formula for each grid.

Now I am going to do a 2 by 3 rectangular grid.

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35 x 47 = 1645

45 x 37 = 1665

Diagonal difference: 20

Now in algebra:

x

x+1

x+2

x+10

x+11

x+12

(x+2) (x+10) – x(x+12)

= x2+2x+10x+20-(x2+12)

= x2+12x+20-x2-12x

= 20

Now I am going to try a 2 by 4 rectangular grid. I predict that the difference would be 30.

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