Ma1: Number Grid
Introduction
My task is to find a formula for any given number grid, I will then go on to find the formula for the squares and then change the squares to rectangles! I have been given a 10 x 10 number grid to start with. I will draw a square around four numbers. I will identify the different squares by using the top left number as the square number. Once I have found my square I will find the product of the top left and bottom right numbers, and then find the different of this from the bottom left and top right's product.
This would be Square Number 1.
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2
x12 = 12
2x11 = 22
22-12 = 10
Grid One
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Results
Square Number 12 = 12x23 = 276
13x22 = 286
Difference = 286-276 = 10
Square number 18 = 18x29 = 522
19x28 = 532
Difference = 532-522 = 10
Square number 45 = 45x56 = 2520
55x46 = 2530
Difference = 2530-2520 = 10
Square number 72 = 72x83 = 5976
82x73 = 5986
Difference = 5986-5976 = 10
Square number 78 = 78x89 = 6942
79x88 = 6952
Difference = 6952-6942 = 10
Table of Results
Square Number
Difference
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8
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45
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72
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78
0
My table of results show that the differences of each square on the 10x10 grid are all 10.
Prediction
I predict that square number 5 will have a difference of 10.
5x16 = 80
6x15 = 90
Difference = 90-80 = 10
Square Number
Difference
5
0
I will now try this with an 8x8 grid, to help with finding the formula for any size grid.
Grid Two
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Results
Square 1 = 1x10 = 10
2x9 = 18
Difference = 18-10 = 8
Square 7 = 7x16 = 112
8x15 = 120
Difference = 120-112 = 8
Square 19 = 19x28 = 532
20x27 = 540
Difference = 540-532 = 8
Square 37 = 37x46 = 1702
38x45 = 1710
Difference = 1710-1702 = 8
Square 49 = 49x58 = 2842
50x57 = 2850
Difference = 2850-2842 = 8
Table of Results
Square Number
Difference
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37
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49
8
My table of results show that the differences of each square on the 8x8 grid are all 8.
Prediction
I predict that Square 55's difference will be 8
55x64 = 3520
56x63 = 3528
Difference = 3528-3520 = 8
Square Number
Difference
55
8
The Formula
I think that I have found the formula for any number grid. I have found from each of my table of results that the difference equals the same as the grid width. For example, my first grid had a width of 10 numbers and each difference of my products was 10. Then for my second grid, there was a grid width of 8 and each difference was 8.
From this I have come up with a formula:
G=D
If I let my grid width equal G and my difference equal D, the above formula should be correct.
Prediction
I predict, using my formula, that a grid width of 6 should have a difference of 6.
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This is a preview of the whole essay
From this I have come up with a formula:
G=D
If I let my grid width equal G and my difference equal D, the above formula should be correct.
Prediction
I predict, using my formula, that a grid width of 6 should have a difference of 6.
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Results
Square 1 = 1x8= 8
2x7 = 14
Difference = 14-8 = 6
Square 5 = 5x12 = 60
6x11 = 66
Difference = 66-60 = 6
Square 15 = 15x22 = 330
16x21 = 336
Difference = 336-330 = 6
Square 25 = 25x32 = 800
26x31 = 806
Difference = 806-800 = 6
Square 29 = 29x36 = 1044
30x35 = 1050
Difference = 1050-1044 = 6
I have now proved that my formula is correct.
If I let D equal difference and G equal grid width, then my formula to know the difference of products on any size grid will be:
G=D
Extension
For this part of my course work, I am going to change the size of the squares and see if I can find a formula that coincides with this change. I will first change the squares to 3x3, drawing around 9 numbers. Then 4x4, drawing around 16 numbers, and if I need to I will also draw around 25 numbers, 5x5.
3X3 Square
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Results
Square 1 = 1x23 = 23
21x3 = 63
Difference = 63-23 = 40
Square 8 = 8x30 = 240
10x28 = 280
Difference = 280-240 = 40
Square 25 = 25x47 = 1175
27x45 = 1215
Difference = 1215-1175 = 40
Square 51 = 51x73 = 3723
27x45 = 3763
Difference = 3763-3723 = 40
Square 77 = 77x99 = 7623
79x97 = 7663
Difference = 7663-7623 = 40
Table of results
Square Number
Difference
40
8
40
25
40
51
40
77
40
My table of results show that each difference of my products came to 40!
Prediction
I predict that square number 48's difference will be 40.
48x70 = 3360
50x68 = 3400
Difference = 3400-3360 = 40
Square Number
Difference
48
40
My prediction was right, the difference did come to 40!
4x4 Squares
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Results
Square 1 = 1x34 = 34
4x31 = 124
Difference = 124-34 = 90
Square 16 = 16x49 = 784
19x46 = 874
Difference = 87-784 = 90
Square 33 = 33x66 = 2178
63x36 = 2268
Difference = 2268-2178 = 90
Square 47 = 47x80 = 3760
50x77 = 3850
Difference = 3850-3760 = 90
Table of results
Square Number
Difference
90
6
90
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90
47
90
My table of results shows that each difference of my products came to 90!
Prediction
For square number 65 I predict that the difference of the two products will be 90
65x98 = 6370
68x95 = 6460
Difference = 6460-6370 = 90
Square Number
Difference
65
90
My prediction was right the difference was 90!
5x5 Squares
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Results
Square 1 = 1x45 = 45
41x5 = 205
Difference = 205-45 = 160
Square 6 = 6x50 = 300
10x46 = 460
Difference = 460-300 = 160
Square 25 = 25x69 = 1725
29x65 = 1885
Difference = 1885-1725 = 160
Square 32 = 32x76 = 2432
36x72 = 2592
Difference = 2592-2432 = 160
Square 51 = 51x95 = 4845
55x95 = 5005
Difference = 5005-4845 = 160
Table of results
Square Number
Difference
60
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60
25
60
32
60
51
60
My table of results shows that each difference of my products came to 160!
Prediction
For square number 56 I predict that the difference of the two products will be 90
56x100 = 5600
60x96 = 5760
Difference = 5760-5600 = 160
Square Number
Difference
56
60
My prediction was right the difference was 160!
Formula
I now think I have enough results to find the formula for the size of the square used in the grid.
Square Size
Difference
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0
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40
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90
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60
From this table, I can see that the differences are all square numbers multiplied by 10.
Here is what I have found:
2x2 Square = 1x1x10
3x3 Square = 2x2x10
4x4 Square = 3x3x10
5x5 Square = 4x4x10
There for the formula for the square would be:
2X2 square = (2-1)²x10
3x3 Square = (3-1)²x10
4x4 Square = (4-1)²x10
5x5 Square = (5-1)²x10
Therefore, if I let S equal the length of the square, and D equal the difference, I think the formula will be:
D = (S-1)²x10
Prediction
For a 10x10 grid I have come up with the formula D = (S-1)²x10, I will now try this for a 6x6 square.
If my formula is correct then the difference of a 6x6 square should be:
D = (6-1)²x10
D = 250
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Results
Square 1 = 1x56 = 56
6x51 = 306
Difference = 306-56 = 250
Square 5 = 5x60 = 300
10x55 = 550
Difference = 550-300 = 250
Square 41 = 41x96 = 3936
46x91 = 4186
Difference = 4186-3936 = 250
Square 45 = 45x100 = 4500
50x95 = 4750
Difference = 4750-4500 = 250
Square Size
Difference
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250
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250
The table shows that my prediction was right.
To enable myself to get a formula for any grid width I will have to think about what part of my proven formula may apply to the this aspect of my investigation.
The formula for a 10x10 grid is:
D = (S-1)²x10
I think that the multiply by 10 part of my proven formula may be to do with the grid, because the grid was 10 wide. So with a grid 8x8 it should be multiply by 8 and with a 6x6 grid you should multiply by 6.
Therefore if I try this with an 8x8 grid, with a 5x5 square, then the difference should be:
D = (S-1)²x8
D = (5-1)²x8
D = 4²x8
D = 16x8
D = 128
If I also try this with a 6x6, grid and a 3x3 square then I should get the difference to be:
D = (S-1)²x6
D = (3-1)²x6
D = 2²x6
D = 4x6
D = 24
Grid one - 8x8 Grid, 5x5 Squares
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Results
Square 1 = 1x37 = 37
5x33 = 165
Difference = 165 - 37 = 128
Square 4 = 4x40 = 160
8x36 = 288
Difference = 288-160 = 128
Square 25 = 25x61 = 1525
29x57 = 1653
Difference = 1653-1525 = 128
Square 28 = 28x64 = 1792
32x60 = 1920
Difference = 1920-1792 = 128
Grid two - 6x6 Grid, 3x3 Squares
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Results
Square 1 = 1x15 = 15
3x13 = 39
Difference = 39-15 = 24
Square 4 = 4x18 = 72
6x16 = 96
Difference = 96-72 = 24
Square 19 = 19x33 = 627
21x31 = 651
Difference = 651-627 = 24
Square 22 = 22x36 = 792
24x34 = 816
Difference = 816-792 = 24
Grid Reference
Predicted Difference
Difference
Grid one - 8x8 Grid, 5x5 Squares
28
28
Grid two - 6x6 Grid, 3x3 Squares
24
24
The table shows that my predictions where right!
I know have enough information to form a formula for any sized square on and sized grid.
If I had a square SxS on a grid W wide then my formula would be:
D=(S-1)²xW
D = Difference
Extension
I am now going to find a formula for a rectangle any size and a grid any size!
I already know how the grid size changes but the rectangle part is a little harder.
After thinking for a while I have realised that my formula would have to change here:
D=(S-1)²xW
I have worked out that this is for the squares because the S equals the square length/ width. If it was not written as a square number it would be written as:
D=(S-1)x(L-1)xW
Note:
D = difference
S = Square Width
L = Square Length
W = Grid Width
Prediction
Therefore, my prediction is that the formula for a rectangle and size on a grid any size will be:
D=(L-1)x(W-1)xG
Note:
D = Difference
L = Rectangle Length
W = Rectangle Width
G = Grid Width
I will check my prediction on a 10x10 grid with a 2x4 rectangle. If my prediction is right then I should get the following results:
D=(L-1)x(W-1)xG
D=(2-1)x(4-1)x10
D=1x3x10
D=30
Then I will also check my prediction on a 8x8 grid with a 3x5 rectangle. I should get the following results if my prediction is correct:
D=(L-1)x(W-1)xG
D=(3-1)x(5-1)x8
D=2x4x8
D=64
Grid one - 10x10 Grid, 2x4 rectangles
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Results
Square 1 = 1x32 = 32
2x31 = 62
Difference = 62-32 = 30
Square 9 = 9x40 = 360
10x39 = 390
Difference = 390 - 360 = 30
Square 43 = 43x74 = 3182
44x73 = 3212
Difference = 3212-3182 = 30
Square 47 = 47x78 = 3666
48x77 = 3696
Difference = 3696-3666 = 30
Grid two - 8x8 Grid, 3x5 rectangles
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Results
Square 1 = 1x35 = 35
3x33 = 99
Difference = 99-35 = 64
Square 5 = 5x39 = 195
7x37 = 159
Difference = 195-159 = 64
Square 10 = 10x44 = 440
12x42 = 504
Difference = 504-440 = 64
Square 14 = 14x48 = 672
16x46 = 736
Difference = 736-672 = 64
From my results, I can confirm that my predicted formula was correct.
Grid Reference
Predicted Difference
Difference
Grid one - 10x10 Grid, 3x4 Rectangles
30
30
Grid two - 8x8 Grid, 3x5 rectangles
64
64
I have now proven that for a grid any size with a rectangle any size the formula will be:
D=(S-1)x(L-1)xW
Formulas
Any Number Grid:
If I let D equal difference and G equal grid width, then my formula to know the difference of products on any size grid will be:
G=D
Square Any Size, Grid Any Size:
If I had a square SxS on a grid W wide, and let D equal difference, then my formula would be:
D=(S-1)²xW
Rectangle Any Size, Grid Any Size:
If I had a rectangle LxW on a grid G wide, and let the difference equal D, then my formula would be:
D=(L-1)x(W-1)xG
Leander Wilson Page 1 08/05/2007
