Number grid

Top Left hand corner x bottom right hand corner = n(n+22) = n? + 22n

Top right hand corner x bottom left hand corner = (n+20)(n+2) = n?+40+22n

(n?+40+22n) - (n? + 22n) = 40

Therefore the difference between the corners multiplied together will always be 40.

I now feel it is time to look at a 4x4 number square on a 100 grid. I will take a 4x4 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.

Example

33

34

35

36

43

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53

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55

56

63

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65

66

33 x 66 = 2178

36 x 63 = 2268

Difference of 90

It is now clear that in a 4 x 4 square the difference will always be 90.

Algebra

I will assign a letter to the first number in the 4x4 square, n. The right hand top corner will therefore be n+3 The left hand bottom corner will then be n+30 The corner diagonally across from it will be n+33

I will then multiply the corner numbers, as shown in the above example.

Top Left hand corner x bottom right hand corner = n(n+33) = n? + 33n

Top right hand corner x bottom left hand corner = (n+30)(n+3) = n?+33n + 90

(n?+33n + 90) - (n? + 33n) = 90

Therefore the difference between the multiplied corner numbers will constantly be 90.

Analyzing the relationship between the differences of the corners multiplied together in different size squares - employing the use of a table to compare the findings.

Size of square

Difference between corner numbers

Difference between differences

2 x 2

0

30

3 x 3

40

50

4 x 4

90

I predict that a 5 x 5 square will have a difference of 160 because the differences of the differences increase by 20 each time e.g. from 30 to 50. Therefore it would make sense for 50 to go up to 70, add this onto the 90 which makes 160, which is my prediction for the 5 x 5 square.

Proving the prediction:

56

57

58

59

60

66

67

68

69

70

76

77

78

79

80

86

87

88

89

90

96

97

98

99

00

56 x 100 = 5600

60 x 96 = 6760

Difference of 160.

Working out a general equation for the difference between the two corners multiplied together with any number of sides on the square

Considering all that the 10 times table contained all of the numbers (10, 40, 90, 160) then I will assume the equation has a x10 in it, and I will therefore remove the 0 from each of the numbers. This leaves 1, 4, 9, and 16. These are all square numbers. It is noticeable that the numbers are the square of the side of square minus 1. This means that b-12 makes sense. I will encompass this with the x10 so it becomes:

0(b-1)2

Testing the equation

I am going to use the example of a 4 x 4 square; I am already aware that the difference is 90.

0(b-1)2 = 10(4-1)2

0 x 32

0 x 9 = 90

This is the correct answer, my equation is accurate.

Working out the same equation through looking at general cases.

I am going to look at the numbers in the corners and how they relate to the width of a side of the number square. When n is the number in the top left hand corner and b is the length of the square.

The top left hand corner is n, so therefore the top right hand corner will be n + the width of the box minus 1, this is because you move along the width of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the box will have plus n + 10 because as you move down vertically the number increases by 10 each time. This will be multiplied by the side of the square minus 1, n+10(b-1). This is because you move down the width of the square, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+10(b-1) of the left hand bottom corner and then add b-1. I will then multiply the corners out, as I do on my investigation.

Top left hand corner x bottom right corner

n x [n +10 (b-1) + (b-1)] = n2 + 10n(b-1) + n(b-1)

= n2 + 11n(b-1)

Right hand top corner x bottom left corner

b-1 = a

[n+(b-1)] [n+10(b-1)]

As both formula have b-1 in I will replace this with an a

(n+a)(n+10a) = n2 + 10a2 + an + 10an

= n2 + 10a2 + 11an

I will now substitute the b-1 back in, instead of the a

= n2 + 10(b-1)2 + 11n(b-1)

The formula has worked for every example; therefore I think it is right. I am now going to look at a different sized grid.

Extension

I have decided to experiment with an 11 by 11 grid for my extension work.

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0

1

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9

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Example1

70

71

81

82

70 x 82 = 5740

71 x 81 = 5751

Difference of 11

Example 2

43

44

54

55

43 x 55 = 2365

44 x 54 = 2376

Difference of 11

It is clear that in a 4 x 4 box, in an 11 x 11 grid, the difference will always be 11.

Conclusion

I conclude that in this investigation, I have created numerous formulas which demonstrated the differences between corner numbers, and how they relate to the side numbers in the box. I have also changed the size of the original grid, but I was too short of time to create a formula for a 4 x 4 box in a 11 x 11 grid, which I would have done, had I worked on this investigation over a lengthier period.

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