Number grid Investigation

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

33 x 88 = 2904

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

83 x 38 = 3154        

3154 – 2904 = 250

Then I found the difference of 250

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

5 x 60 = 300

55 x 10 = 550

550 – 300 = 250

The difference is 250.

41 x 96 = 3936

 91 x 46 = 4186

4186 – 3936 = 250

The difference is still 250.

45 x 100 = 4500

95 x 50 = 4750

4750 – 4500 = 250

The difference is always 250 in a 6x6 box

After investigating up to 6x6 boxes in a 10 x 10 grid I noticed that there was a pattern:

Finding out the formula:

Now I am going to put my results in a table so I can find out the formula.

From this table I will use quadratics to find out the differences as shown below:

1.                 1st diff

2.       10

3.       40        30        2nd diff

4.       90        50        20

5.       160       70        20

6.       250      90        20

I will use these differences into my quadratics equation.

an² + bn + c

I will only use the first part of the formula.

So...   an²= 10n²

Because the 2nd difference 20, is ÷ by 2 so instead of a I will replaced with 10.

The second part of my formula:

Firstly I will add 1 to my equation because my final formula has got something to do with 1. 10 x 1 = 10

10 – 10 (1)²=10

But I have to add my pattern number so my answer will still remain 10.

So its 10 (2-1)²=10

So my formula is:

10(n-1)²     n being the square box number

Now I will test my formula and see if it is correct.

I will test my formula with the box numbers I have already done.

For the 4x4 box which has a difference of 90

10 (4-1)²

10 (3)²= 90

The formula is correct.

To prove again that my formula works with any number of squares I will make a prediction that 7 x 7 box on a 10 x 10 grid will equals 360.

Let’s try it out using my formula.

10 (7-1)²

10 (6)²=360

I got the formula of 10 (n-1)². This formula only works on a 10x 10 grid, for a 9 X 9 grid the formula should be 9 (n-1)2 = the sequence value. I will know test the formula to see if it is correct.

Using my formula I predict that in a 9 by 9 grid a box of 6 x 6 will have a difference of 225.

9(6-1)²=

9(5)²= 225        my prediction is right

 

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

1 x 11 = 11        

I then repeated this with the top right number and the

bottom left number:        

2 x 10 = 20

20 – 11 = 9

Then I found the difference of 9:

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

12 x 22 = 264

21 x 13 = 273        

273 – 264 = 9

The difference is 9.

62 x 72 = 4464

63 x 71 = 4473

4473 – 4464 = 9

The difference is still 9.

34 x 44 = 1496        

43 x 35 = 1505

1505 – 1496 = 9

The difference is always 9 in a 2x2 grid.

After proving and verifying 4 times that 2x2 box in a 9x9 grid difference is 9, 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.

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

7 x 27 = 189        

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

9 x 25 = 225

225 – 189 = 36

Then I found the difference of 36

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

30 x 50 = 1500

48 x 32 = 1536

1536 – 1500 = 36

The difference is 36.

57 x 77 = 4389

75 x 59 = 4425

4425 – 4389 = 36

The difference is still 36.

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 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 x 35 = 175

32 x 8 = 256        

256 – 175 = 81

The difference is 81.

40 x 70 = 2800

67 x 43 = 2881        

2881 – 2800 = 81

The difference is still 81.

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        

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 x 79 = 3081

75 x 43 = 3225

3225 – 3081 = 144

The difference is 144.

1 x 41 = 41

37 x 5 = 185

185 – 41 = 144

The difference is still 144.

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.

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.

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

28 x 78 = 2184        

73 x 33 = 2409

2409 – 2184 = 225

The difference is 225.

3 x 53 = 159

48 x 8 = 384

384 – 159 = 225

The difference is still 225.

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.  

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

        

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        

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.

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

144 x 130 = 18720

129 x 1456 = 18705

18720 – 18705 = 15

The difference is 15.

161 x 175 = 28175

160 x 176 = 28160

18175 – 28160 = 15

The difference is still 15.

113 x 99 = 11187        

114 x 98 = 11172

11187 - 11172 = 15

The difference is always 15 in a 2 by 2 box.

After proving and verifying 4 times that 2x2 box in a 15x15 grid difference is 15, 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 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.

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

199 x 171 = 34029        

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

169 x 201 = 33969

34029 – 33969 = 60

Then I found the difference of 60.

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

69 x 41 = 2829

39 x 71 = 2769

2829 – 2769 = 60

The difference is 60.

34 x 6 = 204

36 x 4 = 144

204 – 144 = 60

The difference is still 60.

58 x 30 = 1740

28 x 60 = 1680        

1740 – 1680 = 60

The difference is always 60 in a 3x3 box.

My prediction for using a 5 x 5 box in a 15 x 15 grid is 240

Now I will use my formula to see if my prediction is right.

15 (5-1)²

15 (4)²= 240

My prediction was right and my formula works on any number of different grids.  

2x2 = 15        45

3x3 = 60                30

4x4 = 135          75

5x5 = 240        105        30

Instead of square boxes I will now look at rectangles and see if my formula will work or maybe there is a new formula.

On a 10 by 10 Grid I will draw rectangles instead of squares.

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

4 x 16 = 64        

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

14 x 6 = 84         

84 – 64 = 20

Then I found the difference of 20.

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

7 x 19 = 133

17 x 9 = 153

153 – 133 = 20

The difference is 20.

88 x 100 = 8800        

98 x 90 = 8820

8820 – 8800 = 20

The difference is still 20.

24 x 36 = 864        

34 x 26 = 884

884 – 864 = 20

The difference is always 20 in a 2 by 3 rectangle.

After proving four times that 2 by 3 rectangle has a difference of 20, I will now try 3 by 4 rectangles and find the difference using the same procedure.

I drew a rectangle around twelve numbers and then found the product of the top left number:

7 x 30 = 210        

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

27 x 16 = 270         

270 – 210 = 60

Then I found the difference of 60.

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

46 x 69 = 3174

66 x 49 = 3234

3234 – 3174 = 60

The difference is 60.

77 x 100 = 7700

97 x 80 = 7760

7760 – 7700 = 60

The difference is still 60.

94 x 71 = 6674

91 x 74 = 6734

6734 – 6674 = 60

The difference is always 60 in a 3 by 4 rectangle.

After proving four times that 3 by 4 rectangle has a difference of 60, I will now try 4 by 5 rectangles and find the difference using the same procedure.

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

1 x 35 = 35         

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

5 x 31 = 155         

155 – 35 = 120

Then I found the difference of 120.

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

41 x 75 = 3075

45 x 71 = 3195

3195 – 3075 = 120

The difference is 120.

10 x 36 = 360

6 x 40 = 240

360 – 240 = 120

The difference is still 120.

45 x 79 = 3555        

75 x 49 = 3675

3675 – 3555 = 120

The difference is always 120 in a 4 by 5 rectangle.

After proving four times that 4 by 5 rectangle has a difference of 120, I will now try 5 by 6 rectangles and find the difference using the same procedure.

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

21 x 66 = 1386          

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

61 x 26 = 1586         

1586 – 1386 = 200

Then I found the difference of 200.

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

91 x 56 = 5096

51 x 96 = 4896

5096 – 4896 = 200

The difference is 200.

84 x 49 = 4116

44 x 89 = 3916

4119 – 3916 = 200

The difference is still 200.

55 x 100 = 5500        

95 x 60 = 5700

5700 – 5500 = 200

The difference is always 200 in a 5 by 6 rectangle.

Finding out the formula:

Now I am going to put my results in a table so I can find out the formula.

1.                 1st diff

2.       20

3.       60          40        2nd diff

4.       120        60        20

5.       200       80        20

My first formula was 10(n-1)²= 10 (n-1) (n-1), this is when I had the same depth and same width because it was a square box. Now that I have different depth and different width tells me that there are two parts in my formula. So instead of 10(n-1)(n-1) the formula should have a different on one of them instead of n.

So the formula should look like this:

10 (n-1) (d-1)

N being the depth of the square and

D being the width of the square.

I predict that a 6 x 7 rectangle will have a difference of 300.

I will now check my formula and see if it works.

10(6-1)(7-1)

10x5x6 = 300

My prediction is right and my formula works.  

So far I have changed:

1. the size of the box: 2 x 2, 3 x 3 etc
2. the size of the grid: 10 by 10, 9 by 9 and 15 by 15
3. the shape of the box: rectangle and square box

Algebra Proof:

On a 15 by 15 grid I will use algebra to find the difference For the 9 by 9 square highlighted:

 X (x+128) = x² + 128x = 128x²

(x+8)(x+120) = x²+ 120x + 8x + 960 =

128x ² + 960

128 x² - 128x² = 960 = 960

Justifying my results:

15

60         45        

135        75        30

240       105    30

We can see that 30 is added onto the first difference every time.

I can now calculate:

105 + 30 = 135

240 + 135 = 375

I predict that in a 6 x 6 square, the product difference will be 200.

Let’s test it:

1 x 46 = 46

6 x 41 = 246

246 – 46 = 200            My prediction was right.

I will now test my formula for rectangles which is 10 (n-1) (d-1) if it will work in a rectangle of multiples of 2.

The formula is:

Z (n-1) (d-1)

Z being the width of the grid

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.

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) (

4-1) = 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