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Introduction

Number Grid

For this task I will first be looking at a number grid from 1 to 100, like the one below:

 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 will start my investigation by looking at 2 by 2 squares. I will draw a square around 4 numbers, find the product of the top left and bottom right numbers and the product of the bottom left and top right numbers, then calculate the difference the between the 2 products. I will see if there are any patterns and if so I will try to work them out algebraically.

I will then look at changing the size of the squares to see if there are any patterns.  I will try looking at 3 by 3 squares, 4 by 4 squares and 5 by 5 squares; I will do the same with these squares as I have with the 2 by 2 squares, I will find the products of the top left and bottom right and the bottom left and top right numbers then calculate the difference between them.

Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles.  I will start by looking at a 2 by 3 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern.  I will then try changing the size of the rectangles and looking for patterns there.  I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles.

I will also look at changing the size of the number grids to see if this has an affect on the patterns.  I will look at a 9 by 9 grid, an 11 by 11 grid and a 5 by 5 grid.  I will be looking for patterns in 2 by 2 squares within the different size grids and trying to find an algebraic formula to explain my findings.

## 2 X 2 Grid

Middle

85

86

87

88

89

90

93

94

95

96

97

98

99

100

23 x 100 = 2300

30 x 93   = 2790

2790 – 2300 = 490

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an      8 X 8 grid is 490. This proves that my formula is correct. But my formula wont work for a 1 X 1 grid or any grid that goes out of the 10 X 10 grid.

Rectangles

Now I a going to try and find the formula for the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in a rectangle. I will do this by finding the formula for changing one side, then I will find the formula for changing the other side and combine the two together.

First I will look at 2 X 3 grids within a 10 X 10 square. Also I will work out the differences algebraically.

Here is an algebraic grid for a 2 X 3 grid within a 10 X 10 grid.

 a a+2 a+10 a+12

Therefore:

a(a + 12) = a² + 12a

(a + 2)(a + 10) = a² + 12a + 20

(a² + 12a + 20) – (a² + 12a) = 20

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 3 grid inside a 10 X 10 grid will always be 20.

Now I will draw a 2 X 3 grid to test if my theory is correct.

 53 54 55 63 64 65

53 x 65 = 3445

55 x 63 = 3465

3465 – 3445 = 20

I have tested my theoryby using numbers. The difference is 20 so therefore it looks like my theory is right.

The Difference for any 2 X 3 grid is always 20 within a 10 X 10 grid.

2 X 4 Grid

Here is an algebraic grid for a 2 X 4 grid within a 10 X 10 grid.

 a a+3 a+10 a+13

Therefore:

a(a + 13) = a² + 13a

(a + 3)(a + 10) = a² + 13a + 30

(a² + 13a + 30) – (a² + 13a) = 30

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 4 grid inside a 10 X 10 grid will always be 30.

Now I will draw a 2 X 4 grid to test if my theory is correct.

 71 72 73 74 81 82 83 84

71 x 84 = 5964

74 x 81 = 5994

5994 – 5964 = 30

I have tested my theoryby using numbers. The difference is 30 so therefore it looks like my theory is right.

The Difference for any 2 X 4 grid is always 30 within a 10 X 10 grid.

2 X 5 Grid

Here is an algebraic grid for a 2 X 5 grid within a 10 X 10 grid.

 a a+4 a+10 a+14

Therefore:

a(a + 14) = a² 14a

(a + 4)(a + 10) = a² + 14a + 40

(a² + 14a + 40) – (a² 14a) = 40

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 5 grid inside a 10 X 10 grid will always be 40.

Now I will draw a 2 X 5 grid to test if my theory is correct.

 86 87 88 89 90 96 97 98 99 100

86 x 100 = 8600

90 x 96   = 8640

8640 – 8600 = 40

I have tested my theoryby using numbers. The difference is 40 so therefore it looks like my theory is right.

The Difference for any 2 X 5 grid is always 40 within a 10 X 10 grid.

2 X 6 Grid

Here is an algebraic grid for a 2 X 5 grid within a 10 X 10 grid.

 a a+5 a+10 a+15

Therefore:

a(a + 15) = a² + 15a

(a + 5)(a + 10) = a² + 15a + 50

(a² + 15a + 50) – (a² + 15a) = 50

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 6 grid inside a 10 X 10 grid will always be 50.

Now I will draw a 2 X 6 grid to test if my theory is correct.

 13 14 15 16 17 18 23 24 25 26 27 28

13 x 28 = 364

18 x 23 = 414

414 – 364 = 50

I have tested my theoryby using numbers. The difference is 50 so therefore it looks like my theory is right.

The Difference for any 2 X 6 grid is always 50 within a 10 X 10 grid.

Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the length but keep the width the same.

Results

 Length Of Rectangle (L) Difference (d) 3 20 4 30 5 40 6 50

After looking at my table I have found out that the difference is 10 subtracted off 10 multiplied the length of the rectangle. For example 5 x 10 = 50, 50 - 10 = 40 and the difference when the length of the rectangle is 5 is 40.

Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the length but keep the width the same is:

10n – 10 simplified to;

d = 10(L – 1)

Now I will test my formula to see if it works.I am going to use an 2 X 7 grid to test to see if my formula works.

d = 10(l – 1)

d = 10(7 – 1)

d = 10 x 6

d = 60

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an      2 X 7 grid should be 60

 51 52 53 54 55 56 57 61 62 63 64 65 66 67

51 x 67 = 3417

57 x 61 = 3477

3477 – 3417 = 60

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in a        2 X 7 grid is 60. This proves that my formula is correct. But my formula wont work for a 1 X 2, a 2 X 1 grid or any grid that goes out of the 10 X 10 grid.

Now I will find the formula for rectangles where I change the width and keep the length the same.

3 X 2 Grid

Here is an algebraic grid for a 3 X 2 grid within a 10 X 10 grid.

 a a+1 a+20 a+21

Therefore:

a(a + 21) = a² + 21a

(a +1)(a + 20) = a² + 21a + 20

(a² + 21a + 20) – (a² + 21a) = 20

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 2 grid inside a 10 X 10 grid will always be 20.

Now I will draw a 3 X 2 grid to test if my theory is correct.

 22 23 32 33 42 43

22 x 43 = 946

23 x 42 = 966

966 – 946 = 20

I have tested my theoryby using numbers. The difference is 20 so therefore it looks like my theory is right.

The Difference for any 3 X 2 grid is always 20 within a 10 X 10 grid.

4 X 2 Grid

 a a+1 a+30 a+31

Therefore:

a(a + 31) = a² + 31a

(a + 1)(a + 30) = a² + 31a + 30

(a² + 31a + 30) – (a² + 31a) = 30

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 2 grid inside a 10 X 10 grid will always be 30.

Now I will draw a 4 X 2 grid to test if my theory is correct.

 6 7 16 17 26 27 36 37

6 x 37 = 222

7 x 36 = 252

253 – 222 = 30

I have tested my theoryby using numbers. The difference is 30 so therefore it looks like my theory is right.

The Difference for any 4 X 2 grid is always 30 within a 10 X 10 grid.

5 x 2 grid

 a a+1 a+40 a+41

Therefore:

a(a + 41) = a² + 41a

(a + 1)(a + 40) = a² + 41a + 40

(a² + 41a + 40) – (a² + 41a) = 40

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 2 grid inside a 10 X 10 grid will always be 40.

Now I will draw a 5 X 2 grid to test if my theory is correct.

 47 48 57 58 67 68 77 78 87 88

47 x 88 = 4136

48 x 87 = 4176

4176 – 4136 = 40

I have tested my theoryby using numbers. The difference is 40 so therefore it looks like my theory is right.

The Difference for any 5 X 2 grid is always 40 within a 10 X 10 grid.

6 X 2 Grid

 a a+1 a+50 a+51

Therefore:

a(a + 51) = a² + 51a

(a + 1)(a + 50) = a² + 51a + 50

(a² + 51a + 50) – (a² + 51a) = 50

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 2 grid inside a 10 X 10 grid will always be 40.

Now I will draw a 5 X 2 grid to test if my theory is correct.

 42 43 52 53 62 63 72 73 82 83 92 93

42 x 93 = 3906

43 x 92 = 3956

3956 – 3906 = 50

I have tested my theoryby using numbers. The difference is 50 so therefore it looks like my theory is right.

The Difference for any 6 X 2 grid is always 50 within a 10 X 10 grid.

Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the width but keep the length the same.

Results

 Width Of Rectangle (w) Difference (d) 3 20 4 30 5 40 6 50

After looking at the table I have noticed that the results are exactly the same as the ‘length of rectangle’ table so therefore the formula should be the same, apart from the fact that you have to change the letter ‘l’ to the letter ‘w’.

Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the width but keep the length the same is:

d = 10(w-1)

Now I will test my formula to see if it works.I am going to use an 8 X 2 grid to test to see if my formula works.

d = 10(w – 1)

d =10(8 – 1)

d = 10 x 7

d = 70

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an      8 X 2 grid should be 70

 3 4 13 14 23 24 33 34 43 44 53 54 63 64 73 74

3 x 74 = 222

4 x 73 = 296

296 – 222 = 70

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an      8 X 2 grid is 70. This proves that my formula is correct. But my formula wont work for a 1 X 2, a 2 X 1 grid or any grid that goes out of the 10 X 10 grid.

Now I will combine the ‘length of rectangles’ formula with the ‘width of rectangles’ formula to get a formula that finds any rectangle.

Any Rectangle

To get the formula to find any rectangle I need to combine, the formula that finds the difference when you change length of the rectangle and the formula that finds the difference when you change the width of the rectangle.

Formula for length of rectangle; d = 10(L – 1)

Formula for width of rectangle; d = 10(w – 1)

If I multiply these two formulae together I should get the formula for any rectangle.

Therefore the formula should be:

d = 10(L – 1)(w – 1)

Now I will test my formula to see if it works.I am going to use an 4 X 5 grid to test to see if my formula works.

d = 10(L – 1)(w – 1)

d = 10(5 – 1)(4 – 1)

d = 10 x 4 x 3

d = 120

The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an      4 X 5 grid should be 120

 16 17 18 19 20 26 27 28 29 30 36 37 38 39 40 46 47 48 49 50

Conclusion

Here is the 8 X 8 grid that I will be using.

 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

First I will look at 3 X 3 grids within a 8 X 8 square. Also I will work out the differences algebraically.

Here is an algebraic grid for a 3 X 3 grid within an 8 X 8 grid.

 a a+2 a+16 a+18

a(a + 18) = a² 18a

(a + 2)(a + 16) = a² + 18a + 32

(a² + 18a + 32) – (a² 18a) = 32

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 3 grid inside an 8 X 8 grid will always be 32.

Now I will draw a 3 X 3 grid to test if my theory is correct.

 21 22 23 29 30 31 37 38 39

21 x 39 = 819

23 x 37 = 851

851 – 819 = 32

I have tested my theoryby using numbers. The difference is 32 so therefore it looks like my theory is right.

The Difference for any 3 X 3 grid is always 32 within an 8 X 8 grid.

4 X 4 Grid

 a a+3 a+24 a+27

a(a + 27) = a² + 27a

(a + 3)(a + 24) = a² + 27a + 72

(a² + 27a + 72) – (a² + 27a) = 72

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 4 grid inside an 8 X 8 grid will always be 96.

Now I will draw a 4 X 4 grid to test if my theory is correct.

 4 5 6 7 12 13 14 15 20 21 22 23 28 29 30 31

4 x 31 = 124

7 x 28 = 196

196 – 124 = 72

I have tested my theoryby using numbers. The difference is 72 so therefore it looks like my theory is right.

The Difference for any 4 X 4 grid is always 72 within an 8 X 8 grid.

5 X 5 Grid

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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