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

e.(mc)2

Extracts from this document...

Introduction

GCSE Advanced Mathematics

image00.png

Coursework

Number Grids

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100

Ahmed Nafi


Squares

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

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  • image01.png

Analysis

10 by 10 Grids

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76

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28

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x

y

Difference

D1

D2

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1

0

10

2

2

10

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30

3

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40

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50

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90

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70

5

5

160

20


8 by 8 Grids

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58

65

66

11 by 11 Grids

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12 by 12 Grids

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13 by 13 Grids

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153

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14 by 14 Grids

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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.

image02.png        ……. (i)

The above formula is of quadratic format,image14.png.

...read more.

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

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x

y

Difference

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D2

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0

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x

y

Difference

D1

D2

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a+b+c

3a+b

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4a+2b+c

2a

5a+b

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9a+3b+c

2a

7a+b

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16a+4b+c

2a

9a+b

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25a+4b+c

2a


Substitute 1

  1. image03.png
  1. image04.png
  1. image05.png

Testing

  1. image06.png
  1. image07.png
  • This shows that the formula for any size squares in a 12 by 12 grid is to be: - image08.png.

I can observe from the above that a pattern is forming between the two formulas. In image09.png, 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: -

image10.png

This is when: -

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

Proof Testing

8 by 8 Grids

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11 by 11 Grids

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13 by 13 Grids

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14 by 14 Grids

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Rectangles

Hypothesis

Example

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  • image11.png
...read more.

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: -

image13.png        …… (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

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10 by 10 Grids

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11 by 11 Grids

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12 by 12 Grids

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13 by 13 Grids

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14 by 14 Grids

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...read more.

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