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## Introduction

For my investigation I will be finding out patterns and differences in a number grid.

An example of a number grid that I am going to use for my investigation is shown below.

Columns

Rows

As you can see I will draw a box on the number grid and multiply the top left number with the bottom right number in the box. I will then do the same for the top right and the bottom left number. Once I have the sum for each multiplication I will then find the difference between the two calculations. I will different size of boxes e.g. 2x2 and 3x3; these numbers represent the number of numbers there are across the side of box and on top of the box. An example of this would be 4x4 means there is 4 numbers on the side and 4 numbers on the top as shown below.

Multiply the top left number with the bottom right number then do the same, multiply the bottom left number with the top right number. After this take both calculations and minus them both from each other.

## Plan

For my plan I will firstly find out calculations for 2x2, 3x3, 4x4 and 5x5 squares and find out the differences for each set of boxes. Once I have found out the differences I will then write up a statement and the predictions for each of my 6x6 and 7x7 squares.

After I have done some data calculation to prove my differences are correct I will then move on to working out differences for rectangular shapes for instance 2x3, 2x4, 2x5, 3x4, 3x5, 4x5 after doing this I will then do another set of statements and predictions to work out 2x6, 2x7, 3x6, 3x7, 4x6, 4x7, and 5x6, 5x7 this would be the second part complete of my investigation. After working out all the square boxes The last piece of data I will do after my rectangular boxes will be doing the whole process again but using a different number grid and finding different patterns and differences in that number grid. After I have completed all of my working out and results I should then do a conclusion to sum up what I have found out from my investigation.

Statement

I have found out the differences for the 2x2, 3x3, 4x4 and 5x5 squares and they are shown below. Following in this statement are my prediction and my results for my 6x6 squares and 7x7 squares.

Differences

2x2 3x3 4x4 5x5

10 40 90 16

30 50 70 variable

20 20 difference in between the variable

One pattern I have found is that all the differences shown above are multiples of the 10x table. I worked this out because all the differences end in 0. I can prove this by saying 10x1 = 10, 10x4 = 40, 10x9 = 90 and

10x16 = 160.

I have also worked out an algebraic pattern and formulae to work out these differences as shown below.

The algebraic formulae I worked out is shown on the next line. D means difference

D = 10 (n-1) 2

Below is another way of setting out the differences for the squares. I have set them out in a table and gradually as I go along I will fill in all the blanks boxes and even add a 6x6 column and a 7x7 column.

Prediction

My prediction for my 6x6 and 7x7 is that they will both be a multiple of 10. I guess that the difference for the 6x6 squares will be 250 and the difference for the 7x7 squares will be 360.

Here are some of the results I found for the 6x6 squares as you can see I have done more then one so that I can prove the difference. All the differences are underlined and in bold.

1 x 56 = 56

51 x 6 = 306

306 – 56 = 250

45 x 100 = 4500

95 x 50 = 4750

4750 – 4500 = 250

13 x 68 = 884

18 x 63 = 1134

1134 – 884 = 250

My prediction for the 7x7 squares was 360. Here are some of the data calculations I come up with for 7x7 squares.

1 x 67 = 67

61 x 7 = 427

427 – 67 = 360

34 x 100 = 3400

94 x 40 = 3760

3760 – 3400 = 360

12 x 78 = 936

72 x 18 = 1296

1296 – 936 = 360

This is the difference table below but as you can see I have filled in the differences to the 6x6 and the 7x7 column

Data calculations for 2x2 squares

1 x 12 = 12 2 x 11 = 22

22 – 12 = 10

6 x 17 = 102 16 x 7 = 112

112 – 102 = 10

57 x 68 = 3876 67 x 58 = 3886

3886 – 3876 = 10

45 x 56 = 2520 55 x 46 = 2530

2530 – 2520 = 10

36 x 47 = 1692 46 x 37 = 1702

1702 – 1692 = 10

9 x 20 = 180 19 x 10 = 190

190 – 18- = 10

71 x 82 = 5822 72 x 81 = 5832

5836 – 5822 = 10

83 x 94 = 7802 93 x 94 = 7812

7812 – 7802 = 10

3 x 14 = 42 13 x 4 = 52

52 – 42 = 10

79 x 90 = 7110 89 x 80 = 7120

7120 – 7110 = 10

23 x 34 = 782 33 x 24 = 792

792 – 782 = 10

Data calculation for 3x3 squares

1 x 23 = 23 21 x 3 = 63

63 – 23 = 40

4 x 26 = 104 24 x 6 = 144

144 – 104 = 40

78 x 100 = 7800 98 x 80 = 7840

7840 – 7800 = 40

65 x 87 = 5655 85 x 67 = 5695

5695 – 5655 = 40

15 x 37 = 555 35 x 17 = 595

595 – 555 = 40

8 x 30 = 240 28 x 10 = 280

280 – 240 = 40

52 x 74 = 3848 72 x 54 = 3888

3888 – 3848 = 40

16 x 38 = 608 36 x 18 = 648

648 – 608 = 40

35 x 57 = 1995 55 x 37 = 2035

2035 – 1995 = 40

Data calculation for 4x4 squares

17x 50 = 850 47 x 20 = 940

940 – 850 = 90

1 x 34 = 34 31 x 4 = 124

124 – 34 = 90

24 x 57 = 1368 54 x 27 = 1458

1458 – 1368 = 90

36 x 69 = 2484 66 x 39 = 2574

2574 – 2484 = 90

41 x 74 = 3034 71 x 44 = 3124

3124 – 3034 = 90

67 x 100 = 6700 97 x 70 = 6790

6790 – 6700 = 90

53 x 86 = 4558 83 x 56 = 4648