3x3

Now I am going to see if a 3x3 grid gets the same out come as a 2x2 grid.

I don’t have the same outcome as my last examples but if I divide 28 by 4 I will get 7. This means I have a multiple of 7.

Again I have 28. I have a multiple of 7 again.

Again all the answers are at least a multiple of 7.

4x4

I am going to see what a 4x4 grid gives me and try to relate it to the main grid size.

If I times 7 by 9 we get 63. So again I have a multiple of 7.

Again 63 is a multiple of 7.

Yet again the answer is 63, a multiple of the number 7.

5x5, 6x6 and 7x7

In this part of my course work I will do only one each of 5x5, 6x6 and 7x7 just to see if the outcomes and the relation is the same as the last few examples I have done.

112 is a multiple of 7. If I were to multiply 7 by 16 my answer would be 112.

Again my answer is a multiple of 7. If I divide my answer by 25 I will get 7.

252 is again a multiple of 7. 7x36=252. So far all my answers for a 7x7 main grid size is seven.

### Rectangles

I am now going to do some rectangles to see if I get the same as my last examples. The rectangles I am going to do are 2x3, 2x4 and 3x4

14 is a multiple of 7.

21 is a gain a multiple of 7.

42 is a multiple of 7 as well.

I have the same out come as my last examples.

### Analysis

After doing my main grid size of 7 x 7 I found that all my answers were multiples of 7.

I can see that there are two patterns here with the grid that I have made. I can see that the pattern is between the grid size and the difference.

### Pattern 1

The first pattern that I can see is:

1 3 2

4 5 2

9 7 2

16 9 2

25 11 2

36

I can see that the difference between the differences go up in two’s. This must mean that the answer is a multiple which is true.

### Pattern 2

I have found that if you take the length and the width of the grid size for example 3x3 then subtract the width and the length each by 1, I get 2x2 then x 2x2 by 7 I get the same pattern as in my table.

Eg1,

- 6x6
- - 1 from each of the width and the length
- I get 5x5
- Then I x that by 7
- I get 5x5x7
- Which adds up to what I got in my table.

Eg2,

- 2x4
- - 1 from each of the length and the width
- I get 1x3
- Then I x that by 7
- I get 1x3x7
- Which adds up to the same as my rectangle I did in my 7x7 grid.

If I make:

- N= width
- M= length
- G= main grid size
- D= difference

I feel that I can make a formula

(Width – 1) x (length – 1) x grid size = difference

so………

(N-1) x (m-1) x G = D

I predict that if I were to use this formula I will be able to work out the differences a lot easily without having to multiply then subtract. To see if my hypothesis is correct I will do an 8x8 grid, 9x9 grid and 10x10 grid.

8x8 grid

In this section of my course work in am going to see if my formula works for an 8x8 grid. In the next section I will be doing a 9x9 grid and eventually a 10x10 grid.

2x2

29x36=1044

28x37=1036

1044-1036=8

I have the answer as 8. Which is the main grid size.

3x3

48x62=2976

46x64=2944

2976-2944=32

32 is a multiple of 8. The main grid size.

For these results I will test my formula on them.

(N-1) x (M-1) x G = D

(2-1) x (2-1) x 8 = D

D = 8

1x1x8=8

So in this case the formula works.

(N-1) x (M-1) x G = D

(3-1) x (3-1) x 8 = D

D = 32

2x2x8=32

Again my formula has worked.

I will try some more just to make sure.

4x4

13x34=442

10x37=370

442-370=72

72 is a multiple of 8. if I were to divide 72 by 9 it would equal to 8.

5x5

15x43=645

11x47=517

645-517=128

16x8 = 128. Which is a multiple of 8.

Now it is time to check my formula again.

(N-1) x (M-1) x G = D

(4-1) x (4-1) x 8 = D

D = 72

3x3x8=72

My formula has worked again.

(N-1) x (M-1) x G = D

(5-1) x (5-1) x 8 = D

D = 128

4x4x8=128

Again my formula has worked.

9x9 grid

I will now see if it works with a 9x9 grid this time only with a 2x2 and a 9x9.

2x2

21x29=609

20x30=600

609-600=9

The answer is 9, the main grid size

9x9

9x73=657

1x81=81

657-81=576

576 divided by 9 = 64 64x9= 576 which is the answer for this grid size. It is still a multiple of 9.

It is now time to test my formula with the last 9x9 grids.

(N-1) x (M-1) x G = D

(2-1) x (2-1) x 9 = D

D = 9

1x1x9=9

My formula has worked again.

(N-1) x (M-1) x G = D

(9-1) x (9-1) x 9 = D

D = 648

8x8x9=576

Again my formula has worked.

I don’t think that I have to prove my answers now with a 10x10 grid so I wont. I think you can see that I have succeeded in my course work. I am very happy in my out come and I feel that I have done very well.