# I will attempt to discover a general formula that will find the difference between the product of the top left number and the bottom right and the product of top right and bottom left (diagonals) of any size square and any size grid.

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Introduction

Maths Coursework

In this investigation I will attempt to discover a general formula that will find the difference between the product of the top left number and the bottom right and the product of top right and bottom left (diagonals) of any size square and any size grid.

I will start on a 10 by 10 grid and a square of 4 numbers, by picking a square of 4 numbers and multiplying the top left number with the bottom right and then finding the difference between this and the product of top right and bottom left.

Eg.

3 3 | 4 |

13 13 | 14 |

In this case, it would be the difference between 3 x 14 and 4 x 13, which is 10.

2x2 square

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 |

1 | 2 |

11 | 12 |

1 x 12 = 12

2 x 11 = 22

## Difference between the product of the diagonals is 10

44 | 45 |

54 | 55 |

44 x 55 = 2420

45 x 54 = 2430

Difference = 10

77 | 78 |

87 | 88 |

77 x 88 = 6776

78 x 87 = 6786

Difference = 10

18 | 19 |

28 | 29 |

18 x 29 = 522

19 x 28 = 532

Difference = 10

Pattern: As you can

Middle

121

11 is added as you go vertically down the grid

Here we see that the number added vertically is the width of the grid. As 10 is the width of the grid we can substitute this into the equation by representing the width as w

n | n+1 |

n+w | n+w+1 |

n x (n+w+1) = n2 + nw +n

(n+1) x (n+w) = n2 + nw + n + w

As n2 + nwcancels out,the difference will always be w, which is the width of the grid

## 3x3 square

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 |

1 | 2 | 3 |

11 | 12 | 13 |

21 | 22 | 23 |

1 x 23 = 23

3 x 21 = 63

## Difference = 40

71 | 72 | 73 |

81 | 82 | 83 |

91 | 92 | 93 |

71 x 93 = 6603

73 x 91 = 6643

Difference = 40

78 | 79 | 80 |

88 | 89 | 90 |

98 | 99 | 100 |

78 x 100 = 7800

80 x 98 = 7840

Difference = 40

8 | 9 | 10 |

18 | 19 | 20 |

28 | 29 | 30 |

8 x 30 = 240

10 x 28 = 280

Difference = 40

We can turn the values found in the 3x3 square into albegra:

n | n+2 | |

Dd | ||

n +2w | n +w+2 |

n x (n+2w+2) = n2 + 2nw +2n

(n+2) x (n+2w) = n2 + 2nw + 2n + 4w

Here the difference will always be 4w

4x4 square

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 |

1 | 2 | 3 | 4 |

11 | 12 | 13 | 14 |

21 | 22 | 23 | 24 |

31 | 32 | 33 | 34 |

1 x 34 = 34

4 x 31 = 124

## Difference = 90

61 | 62 | 63 | 64 |

71 | 72 | 73 | 74 |

81 | 82 | 83 | 84 |

91 | 92 | 93 | 94 |

61 x 94 = 5734

64 x 91 = 5824

Difference = 90

67 | 68 | 69 | 70 |

77 | 78 | 79 | 80 |

87 | 88 | 89 | 90 |

97 | 98 | 99 | 100 |

67 x 100 = 6700

70 x 97 = 6790

Difference = 90

7 | 8 | 9 | 10 |

17 | 18 | 19 | 20 |

27 | 28 | 29 | 30 |

37 | 38 | 39 | 40 |

7 x 40 = 280

10 x 37 = 370

Difference = 90

n | 9 | n+3 | |

17 | 19 | 20 | |

27 | 29 | 30 | |

n +3w | 100 | 100 | n +3w+3 |

Conclusion

If we represent the length of the square as l then the number added would be (l –1)

n | n+(l –1) |

n+w | n+w+(l –1) |

n | n+(l –1) | |

Dd | ||

n +(l –1)w | n +w+(l –1) |

n | 9 | n+(l –1) | |

17 | 19 | 20 | |

27 | 29 | 30 | |

n +(l –1) w | 100 | 100 | n +3w+(l –1) |

n (n +3w+(l –1)) = n2 + 3nw + (l –1)n

n+(l –1) x n +(l –1)w = n2 + 3nw + (l –1)n + (l –1) 2w

The difference of any size square of numbers is:

W(l –1)2i.e width of the grid x ((the length of the square - 1) squared)

### Checking the formula

#### With the formula W(l –1)2we should be able to find the difference on any size grid and any size square.

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 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 |

113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 |

127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 |

141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 |

155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 |

169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 | 180 | 181 | 182 |

183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 |

Grid 14x14, square 8x8

By using the formula, the difference should be:

W(l –1)2 = 14 (8-1) 2

= 686

Check:

61x166 = 10126

68x159 = 10812

Difference does equal 686

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 |

Grid 7x7, square 7x7

By using the formula, the difference should be:

W(l –1)2 = 7 (7-1) 2

= 252

Check:

1x49 = 49

7x43 = 301

Difference does equal 252

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

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