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• Level: GCSE
• Subject: Maths
• Word count: 1747

e.(mc)2

Extracts from this document...

Introduction

Coursework

Number Grids

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Ahmed Nafi

Hypothesis

The difference of products of the diagonally opposite numbers will be constant throughout the grid and the difference of 2 by 2 square boxes in any grid will be equal to the grid length.

Example

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

 35 36 45 46

Analysis

10 by 10 Grids

 13 14 23 24
 66 67 76 77
 18 19 28 29
 x y Difference D1 D2 1 1 0 10 2 2 10 20 30 3 3 40 20 50 4 4 90 20 70 5 5 160 20

8 by 8 Grids

 57 58 65 66

11 by 11 Grids

 101 102 112 113

12 by 12 Grids

 42 43 54 55

13 by 13 Grids

 140 141 153 154

14 by 14 Grids

 91 92 105 106

Conclusion

From the analysis, we can observe that the difference of products of the diagonally opposite numbers is always constant, which proved the first part of the hypothesis as true. The difference of 2 by 2 squares for any grid is always equal to the grid-length and thus proves the second part of the hypothesis as true.

From the series of these results, a formula can be developed, as shown below.

……. (i)

The above formula is of quadratic format,.

Middle

As I have found the formula for any square box in a 10 by 10 grid, I will not try and attempt to find a general formula for any size square box for any size square grid. I will do the same to 12 by 12 grids as I did for 10 by 10 grids.

12 by 12 Grids

 19 20 31 32
 89 90 101 102
 14 15 26 27
 x y Difference D1 D2 1 1 0 12 2 2 12 24 36 3 3 48 24 60 4 4 108 24 84 5 5 192 24 x y Difference D1 D2 1 1 a+b+c 3a+b 2 2 4a+2b+c 2a 5a+b 3 3 9a+3b+c 2a 7a+b 4 4 16a+4b+c 2a 9a+b 5 5 25a+4b+c 2a

Substitute 1

Testing

• This shows that the formula for any size squares in a 12 by 12 grid is to be: - .

I can observe from the above that a pattern is forming between the two formulas. In , a and c is always equal to the grid-length and b is always equal to the grid-length multiplied by 2.

Hence, I can deduce a general formula for any size square in any grid size. The formula is: -

This is when: -

• n = grid-length
• x = square length/height

Proof Testing

8 by 8 Grids

 26 27 28 34 35 36 42 43 44

11 by 11 Grids

 16 17 18 19 20 27 28 29 30 31 38 39 40 41 42 49 50 51 52 53 60 61 62 63 64

13 by 13 Grids

 72 73 74 85 86 87 98 99 100

14 by 14 Grids

 11 12 25 26

Hypothesis

Example

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 23 24 25 33 34 35

Conclusion

From the set of these results and the pattern, we can derive a formula to determine the difference of products of the diagonally opposite numbers. The formula is: -

…… (iii)

Where: -

• x = base length of parallelogram
• y = vertical height of parallelogram
• n = grid-length

The formula states that ‘to find the difference of products of the diagonally opposite numbers, we need to multiply the grid-length minus one by the base length minus one by the vertical length minus one’. I have based the formula around the properties of rectangles. This is because the parallelogram is basically a rectangle when the angle between the adjacent sides is not 90°. So what I have done is use the formula (ii) in modified format. That is, by changing the slope of the vertical sides.

Testing

8 by 8 Grids

 3 4 5 6 7 11 12 13 14 15 19 20 21 22 23 27 28 29 30 31

10 by 10 Grids

 1 2 3 4 11 12 13 14 21 22 23 24 31 32 33 34

11 by 11 Grids

 45 46 47 48 49 56 57 58 59 60 67 68 69 70 71 78 79 80 81 82

12 by 12 Grids

 6 7 8 9 10 18 19 20 21 22 30 31 32 33 34 42 43 44 45 46

13 by 13 Grids

 71 72 73 74 75 84 85 86 87 88 97 98 99 100 101 110 111 112 113 114

14 by 14 Grids

 100 101 102 103 104 114 115 116 117 118 128 129 130 131 132 142 143 144 145 146

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