# e.(mc)2

Extracts from this document...

Introduction

GCSE Advanced Mathematics

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

## 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

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 |

## Rectangles

## 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 |

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month