# Number grid.

Extracts from this document...

Introduction

## NUMBER

## GRID

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 |

### Project by: Clare Bray

###### Date completed: May 2003

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.

10 x 10 Grid

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 |

#### 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 and recorded the results in a table:

Box size: 2 x 2

Box | numbers | difference |

1 | 12 x 23 = 276 13 x 22 = 286 | 286 – 276 = 10 |

2 | 27 x 38 = 1026 28 x 37 = 1036 | 1036 – 1026 = 10 |

Box | numbers | difference |

3 | 44 x 55 = 2420 45 x 54 = 2430 | 2430 – 2420 =10 |

4 | 73 x 84 = 6132 74 x 83 = 6142 | 6142 – 6132 = 10 |

When using the 3 x 3 box obviously I was only using the numbers in the four corners of the box thereby omitting five numbers in between i.e.

6 | 7 | 8 |

16 | 17 | 18 |

26 | 27 | 28 |

6 x 28 = 168 208 – 168 = 40

8 x 26 = 208

Middle

11 x 23 = 253

13 x 21 = 273

273 – 253 = 20

2

15 x 27 = 405

17 x 25 = 425

425 – 405 = 20

3

42 x 54 = 2268

44 x 52 = 2288

2288 – 2268 = 20

4

64 x 76 = 4864

66 x 74 = 4884

4884 – 4864 = 20

I therefore chose to repeat this exercise again with a larger rectangle to try to prove my prediction. Following the same action as previously I drew a 4 x 3 rectangle around the numbers and again omitting the numbers in the middle.

14 | 15 | 16 | 17 |

24 | 25 | 26 | 27 |

34 | 35 | 36 | 37 |

Box size: 4 x 3

Box | numbers | Difference |

1 | 14 x 37 = 518 17 x 34 = 578 | 578 – 518 = 60 |

2 | 41 x 64 = 2624 44 x 61 = 2684 | 2684 – 2624 = 60 |

3 | 46 x 69 = 3174 49 x 66 = 3234 | 3234 – 3174 = 60 |

4 | 75 x 98 = 7350 78 x 95 = 7410 | 7410 – 7350 = 60 |

Box number | Box size | Difference |

1 | 3 x 2 | 20 |

2 | 4 x 3 | 60 |

3 | 5 x 4 | 120 |

4 | 6 x 5 | 200 |

To find the difference for box 7 x 6 use box 6 x 5

6 x 5 = 30

30 x 10 = 300

Pattern:

## Box size | 3 x 2 | 4 x 3 | 5 x 4 | 6 x 5 | 7 x 6 |

= | 6 | 12 | 20 | 30 | 42 |

x 10 x 10 x 10 x 10

## Difference | 20 | 60 | 120 | 200 | 300 |

On closer examination, the differences for both box shapes display another pattern. For the square box the difference is: add 30 to the 10 to make the difference 40, then add 20 to the 30 to find the next difference and so on.

Box size | 2 x 2 | 3 x 3 | 4 x 4 | 5 x 5 | 6 x 6 | 7 x 7 |

Difference | 10 | 40 | 90 | 160 | 250 | 360 |

+30 +50 +70 +90 +110

+20 +20 +20 +20

The process to find the difference for the rectangular box is, again, almost the same as for the square box:

Box size | 3 x 2 | 4 x 3 | 5 x 4 | 6 x 5 | 7 x 6 |

Difference | 20 | 60 | 120 | 200 | 300 |

+40 +60 +80 +100

+20 +20 +20

I believe that the patterns are so similar because the grid used was the same for both shaped boxes, I also believe that the number sequences of the differences are multiples of ten because the grid has ten numbers in each row.

Because of this belief I chose to try another number grid, this time I would be using a grid that was 8 x 8.

I will be using the same process as before of finding the product of the top left number and bottom right number, then the same with the top right number and bottom left number. I will then find the difference by subtracting the smaller of the two products from the larger.

8 x 8 Grid

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 |

Box size 2 x 2

Box | numbers | Difference |

1 | 9 x 18 = 162 10 x 17 = 170 | 170 – 162 = 8 |

2 | 21 x 30 = 630 22 x 29 = 638 | 638 – 630 = 8 |

3 | 34 x 43 = 1462 35 x 42 = 1470 | 1470 – 1462 = 8 |

4 | 53 x 62 = 3286 54 x 61 = 3294 | 3294 – 3286 = 8 |

12 | 13 | 14 |

20 | 21 | 22 |

28 | 29 | 30 |

12 x 30 = 360 392 – 360 = 32

14 x 28 = 392

Box size 3 x 3

Box | numbers | Difference |

1 | 12 x 30 = 360 14 x 28 = 392 | 392 – 360 = 32 |

2 | 25 x 43 = 1075 27 = 41 = 1107 | 1107 – 1075 = 32 |

3 | 30 x 48 = 1440 32 x 46 = 1472 | 1472 – 1440 = 32 |

4 | 43 x 61 = 2623 45 x 59 = 2655 | 2655 – 2623 = 32 |

Box number | Box size | Difference |

1 | 2 x 2 | 8 |

2 | 3 x 3 | 32 |

3 | 4 x 4 | 72 |

4 | 5 x 5 | 128 |

2 x 47 = 94 294 – 94 = 200

7 x 42 = 294

Pattern:

## Box size | 2 x 2 | 3 x 3 | 4 x 4 | 5 x 5 | 6 x 6 |

= | 4 | 9 | 16 | 25 | 36 |

x 8 x 8 x 8 x 8

## Difference | 8 | 32 | 72 | 128 | 200 |

Conclusion

On closer inspection of the patterns, I was able to see an easier way to explain how to work out the difference using algebra. By using the tables showing the number patterns, I could draw a simpler version that also enabled me to find the nth term:

#### Number pattern for square box, 10 x 10 grid

Box number (n) | 1 | 2 | 3 | 4 | 5 | 6 |

Difference (d) | 10 | 40 | 90 | 160 | 250 | 360 |

#### Therefore n = box number and d = difference

d = 10n2

Example: d = 10 x 22

= 10 x 4

= 40

Therefore the next term of the sequence would be:

d = 10 x 32

= 10 x 9

= 90

This way to find the way the number pattern works also works for the 8 x 8 grid:

Number pattern for square box, 8 x 8 grid

Box number (n) | 1 | 2 | 3 | 4 | 5 |

Difference (d) | 8 | 32 | 72 | 128 | 200 |

d = 8n2

Therefore: d = 8 x 22

= 8 x 4

= 32

And the next term would be:

d = 8 x 32

= 8 x 9

= 72

As before this same, process applies to the 12 x 12 grid:

Number pattern for square box, 12 x 12 grid

Box number (n) | 1 | 2 | 3 | 4 |

Difference (d) | 12 | 48 | 108 | 192 |

d = 12n2

Therefore: d = 12 x 22

= 12 x 4

= 48

And the next term in the sequence:

d = 12 x 32

= 12 x 9

= 108

The same rule for finding the nth term applies for all number patterns with regards to the square boxes, and, although the patterns for the rectangular boxes are very similar to that of the square boxes, I was unable to find the nth term. I believe that this is because the patterns created by the rectangular boxes are not linear sequences.

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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