# Number grid Investigation

Extracts from this document...

Introduction

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 |

For my coursework on investigation I have chosen to use a number grid. From the grid I will draw a box around a selected amount of numbers I will then find the product of the top left number and the bottom right number this will be repeated with the top right number and the bottom left number. When this is completed I will then find the difference between the two numbers.During this process I will be looking for patterns in and relationships between the differences that I collect.

The grid that I have started with is numbered from 1 to 100 with ten in each row and ten in each column.

12 | 13 |

22 | 23 |

#### I drew a box around four numbers and then found the product of the top left number:

12 x 23 = 276

I then repeated this with the top right number and the bottom left number:

13 x 22 =286

On completion of this I found the difference:

286 - 276 = 10

I repeated this process four times with other numbers from the grid to see if the difference would change.

62 | 63 |

72 | 73 |

62 x 73 = 4526

63 x 72 = 4536

4536 – 4526 = 10

The difference is still 10.

67 | 68 |

77 | 78 |

67 x 78 = 5226

77 x 68 = 5236

5236 – 5226 = 10

The difference is still 10

59 | 60 |

69 | 70 |

59 x 70 = 4130

69 x 60 = 4140

4140 – 4130 = 10

The difference is always 10

After proving and verifying 4 times that 2x2 box in a 10x10 grid difference is 10, I was curious as to what would happen if I changed the size of the box. Therefore I chose to change the box from 2 x 2, (four numbers,) to 3 x 3, (nine numbers,) and used the same process of finding the product of the top left number and bottom right number and vice versa. Then finding the difference.

4 | 5 | 6 |

14 | 15 | 16 |

24 | 25 | 26 |

I drew a box around nine numbers and then found the product of the top left number:

4 x 26 = 104

I then repeated this with the top right number and the bottom left number:

24 x 6 = 144

144 – 104 = 40

Then I found the difference of 40:

Middle

48

49

50

30 x 50 = 1500

48 x 32 = 1536

1536 – 1500 = 36

The difference is 36.

57 | 58 | 59 |

66 | 67 | 68 |

75 | 76 | 77 |

57 x 77 = 4389

75 x 59 = 4425

4425 – 4389 = 36

The difference is still 36.

55 | 56 | 57 |

64 | 65 | 66 |

73 | 74 | 75 |

55 x 75 = 4125

73 x 54 = 3942

4125 – 3942 = 36

The difference is always 36 in a 3 x3 box.

After proving and verifying 4 times that 3x3 box in a 9x9 grid difference is 36, I was curious as to what would happen if I changed the size of the box again. Therefore I chose to change the box from 3 x 3, (nine numbers,) to 4 x 4, (sixteen numbers,) and used the same process of finding the product of the top left number and bottom right number and vice versa. Then finding the difference.

I drew a box around sixteen numbers and then found the product of the top left number:

1 | 2 | 3 | 4 |

10 | 11 | 12 | 13 |

19 | 20 | 21 | 22 |

28 | 29 | 30 | 31 |

1 x 31 = 31

I then repeated this with the top right number and the bottom left number:

28 x 4 = 112

112 – 31 = 81

Then I found the difference of 81

I repeated this process four times with other numbers from the grid to see if the difference would change.

5 | 6 | 7 | 8 |

14 | 15 | 16 | 17 |

23 | 24 | 25 | 26 |

32 | 33 | 34 | 35 |

5 x 35 = 175

32 x 8 = 256

256 – 175 = 81

The difference is 81.

40 | 41 | 42 | 43 |

49 | 50 | 51 | 52 |

58 | 59 | 60 | 61 |

67 | 68 | 69 | 70 |

40 x 70 = 2800

67 x 43 = 2881

2881 – 2800 = 81

The difference is still 81.

46 | 47 | 48 | 49 |

55 | 56 | 57 | 58 |

64 | 65 | 66 | 67 |

73 | 74 | 75 | 76 |

46 x 76 = 3496

73 x 49 = 3577

3577 – 3496= 81

The difference is always 81 in a 4 x 4 box.

After proving and verifying 4 times that 4x4 box in a 9x9 grid difference is 81, I was curious as to what would happen if I changed the size of the box again. Therefore I chose to change the box from 4 x 4, (sixteen numbers,) to 5 x 5, (twenty-five numbers,) and used the same process of finding the product of the top left number and bottom right number and vice versa. Then finding the difference.

I drew a box around twenty-five numbers and then found the product of the top left number:

4 x 44 = 176

4 | 5 | 6 | 7 | 8 |

13 | 14 | 15 | 16 | 17 |

22 | 23 | 24 | 25 | 26 |

31 | 32 | 33 | 34 | 35 |

40 | 41 | 42 | 43 | 44 |

I then repeated this with the top right number and the bottom left number:

40 x 8 = 320

320 – 176 = 144

Then I found the difference of 144.

I repeated this process four times with other numbers from the grid to see if the difference would change.

39 | 40 | 41 | 42 | 43 |

48 | 49 | 50 | 51 | 52 |

57 | 58 | 59 | 60 | 61 |

66 | 67 | 68 | 69 | 70 |

75 | 76 | 77 | 78 | 79 |

39 x 79 = 3081

75 x 43 = 3225

3225 – 3081 = 144

The difference is 144.

1 | 2 | 3 | 4 | 5 |

10 | 11 | 12 | 13 | 14 |

19 | 20 | 21 | 22 | 23 |

28 | 29 | 30 | 31 | 32 |

37 | 38 | 39 | 40 | 41 |

1 x 41 = 41

37 x 5 = 185

185 – 41 = 144

The difference is still 144.

5 | 6 | 7 | 8 | 9 |

14 | 15 | 16 | 17 | 18 |

23 | 24 | 25 | 26 | 27 |

32 | 33 | 34 | 35 | 36 |

41 | 42 | 43 | 44 | 45 |

5 x 45 = 225

41 x 9 = 369

369 – 225 = 144

The difference is always 144 in a 5x5 box.

After proving and verifying 4 times that 5x5 box in a 9x9 grid difference is 144, I was curious as to what would happen if I changed the size of the box again. Therefore I chose to change the box from 5 x 5, (twenty-five numbers,) to6 x 6, (thirty-six numbers,) and used the same process of finding the product of the top left number and bottom right number and vice versa. Then finding the difference.

31 | 32 | 33 | 34 | 35 | 36 |

40 | 41 | 42 | 43 | 44 | 45 |

49 | 50 | 51 | 52 | 53 | 54 |

58 | 59 | 60 | 61 | 62 | 63 |

67 | 68 | 69 | 70 | 71 | 72 |

76 | 77 | 78 | 79 | 80 | 81 |

I drew a box around thirty-six numbers and then found the product of the top left number:

31 x 81 = 2511

I then repeated this with the top right number and the bottom left number:

36 x 76 = 2736

2736 – 2511 = 225

Then I found the difference of 225.

28 | 29 | 30 | 31 | 32 | 33 |

37 | 38 | 39 | 40 | 41 | 42 |

46 | 47 | 48 | 49 | 50 | 51 |

55 | 56 | 57 | 58 | 59 | 60 |

64 | 65 | 66 | 67 | 68 | 69 |

73 | 74 | 75 | 76 | 77 | 78 |

28 x 78 = 2184

73 x 33 = 2409

2409 – 2184 = 225

The difference is 225.

3 | 4 | 5 | 6 | 7 | 8 |

12 | 13 | 14 | 15 | 16 | 17 |

21 | 22 | 23 | 24 | 25 | 26 |

30 | 31 | 32 | 33 | 34 | 35 |

39 | 40 | 41 | 42 | 43 | 44 |

48 | 49 | 50 | 51 | 52 | 53 |

3 x 53 = 159

48 x 8 = 384

384 – 159 = 225

The difference is still 225.

12 | 13 | 14 | 15 | 16 | 17 |

21 | 22 | 23 | 24 | 25 | 26 |

30 | 31 | 32 | 33 | 34 | 35 |

39 | 40 | 41 | 42 | 43 | 44 |

48 | 49 | 50 | 51 | 52 | 53 |

57 | 58 | 59 | 60 | 61 | 62 |

12 x 62 = 744

17 x 57 = 969

969 – 744 = 225

The difference is always 225 in a 6 x 6 box.

My formula worked on my prediction and I got the right difference.

n | 2x2 | 3x3 | 4x4 | 5x5 | 6x6 |

9 | 36 | 81 | 144 | 225 |

Now I will try a 15 by 15 grid and see if the formula will change.

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 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 | 210 |

211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 | 225 |

I will try different square box inside the 15 by 15 grid like I did in the 10 by 10 grids and see if the formula would change or work.

I drew a box around four numbers and then found the product of the top left number:

51 x 37 = 1887

36 | 37 |

51 | 52 |

I then repeated this with the top right number and the bottom left number:

36 x 52 = 1872

1887 – 1872 = 15

Then I found the difference of 15.

129 | 130 |

144 | 145 |

144 x 130 = 18720 ...read more.

Conclusion

N being the width of the square within grid

D being the depth of square.

So……

10(5-1) (3-1) = 80

Using the formula the difference should be 80. lets see if it is correct.

46 x 94 = 4324

54 x 86 = 4644

4644 – 4324 = 320

The formula is not correct so I will try a 4 x 3 square.

88 | 90 | 92 | 94 |

108 | 110 | 112 | 114 |

128 | 130 | 132 | 134 |

Z (n-1) (d-1)

10(4-1) (3-1) = 60

Using the formula the difference should be 80. Let’s see if it is correct. The formulas

88 x 134 = 11792

94 x 128 = 12032

12032 – 11792 = 240

It’s not correct again.

Both times I’ve tested my formula I noticed that the difference is 4 times the amount of the results taken from formula 2.

Therefore the formula must be:

4z (n-1) (d-1)

Let’s try it on a 5 x 4 square.

4 x 10 (5-1) (

4 | 6 | 8 | 10 | 12 |

24 | 26 | 28 | 30 | 32 |

44 | 46 | 48 | 50 | 52 |

64 | 66 | 68 | 70 | 72 |

4-1) = 480

Let’s see if it is correct.

4 x 72 = 288

12 x 64 = 768

768 – 288 = 480

This is correct.

In the one above they were multiples of 2 the formula was 4z(n-1) (d-1) the multiple number 2², is squared to give 4z.

I predict in a grid of 10 by 10 with multiples of 3, the formula will be

9z (n-1) (d-1)

Because 3² is 9

Let’s see if it is correct in a 5 x 4 square.

9 x 10 (5-1) (4-1) = 1080

12 x 114 = 1368

24 x 102 = 2448

2448 – 1368 = 1080

My prediction is right and so is the formula.

In conclusion to my project I think I handled really well. I have presented my results and formulas in an appropriate way. I have challenged my thought by using predictions throughout the experiment and tested them. I have used algebra where necessary to prove why the formulas or calculations have worked out in the way that they have. Unfortunately I have not been able to present my results in any form chart but I have used tables and number patterns to make my results easier to understand. If I were to extend this project further I would have gathered a large set of formulas and results and gone into greater depth with my approaches carried out. This could prove difficulty as presenting it in an interesting way would cause great problems.

By Kristi Sylari

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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