The factors
which I will change is the size of the square or rectangle.
The size of
the grid.
This is
the grid I will use to begin with: a 10x10 grid.
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I have
taken a random 4x4. I am going to multiply the numbers from the
opposite corners of the square, then find the difference.
12 and 42 are
numbers opposite corners.
15 and 45 are
also numbers in opposite corners.
12 x 42 =
540 15 x 42 = 630
630 –
540 = 90
The
difference is 90, I am now going to investigate other 4 x 4 squares.
36 and 69 are
opposite corners, 39 and 66 are also opposite corners.
36 x 69 =
2484 39 x 66 = 2574
2474 –
2484 = 90
4 and 37 are
opposite corners.
7 and 34 are
opposite corners.
61 x
94 = 5734 64 x 91 = 5824
5824 –
5734 = 90
I have tried
3 examples of 4 x 4 squares in a 10 x 10 grid they and the
differences of the products of the opposite corners always seems to
equal 90.
I am going to
try a different sized square in a 10 x 10 grid.
First I will
try a 2 x2 square.
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1
x 12
= 12 2
x 11
= 22
22 –
12 = 10
8
x 19
= 152 9
x 18
= 162
162 –
152 = 10
42
x 53
= 2226 43
x 52
= 2236
2236 –
2226 = 10
I have tried
3 examples of 2 x 2 squares in a 10 x 10 grid the difference is
always 10.
Now I will
try a 3 x 3 square in a 10 x10.
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1
x 23
= 23 3
x 21
= 63
63
– 23
= 40
78
x 100
= 7800 80
x 98
= 7840
7840 –
7840 = 40
8
x 30
= 240 10
x 28
= 280
280 –
240 = 40
The
difference of the products of the opposite corners is always 40 for a
3 x 3 square.
Now I will
try a 5 x 5 square.
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1
x 45
= 45 5
x 41
= 205
205 –
45 = 160
6
x 50
= 300 56
x 10
= 560
560 –
300 = 160
55
x 99
= 5445 59
x 95
= 5605
5605 –
5445 = 160
The
difference of the products of the opposite corners is always 160 in a
5 x 5 square.
Now I will
try a 6 x 6 square.
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1 x 56 =
56 6 x 51 = 306
306 –
56 = 250
25 x 80 =
2000 30 x 75 = 2250
2250 –
2000 = 250
The
difference of the products of the opposite corners of a 6 x 6 is
always 250.
I am
going to put all the results that I have collected in a table.
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Square size
|
Differences
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2 x 2
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10
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3 x 3
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40
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4 x 4
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90
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5 x 5
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160
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6 x 6
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250
|
I have
noticed from the results that the differences are all square numbers
multiplied by 10, eg
1 x 10 = 10
2 x 10 = 40
3 x 10 = 90
4 x 10 = 160
5 x 10 = 250
Also I have
noticed that the square number which is multiplied by 10 is 1 less
than the length of the square, eg
2 – 1 = 1 1
x 10 = 10
3 – 1 = 2 2
x 10 = 40
4 – 1 = 3 3
x 10 = 90
5 – 1 = 4 4
x 10 = 160
6 – 1 = 5 5
x 10 = 250
From
this I can work out a formula for the difference of the products of
the opposite corners of any size square in a 10 x 10 grid.
n = the
length of the square.
Difference
= (n – 1) x 10 or 10(n – 1)
This
formula shows how to find the difference of any products of opposite
corners of any size square in a 10 x 10 grid.
Proof
I will use n
to represent the original number in top left of the square inside the
grid. It is always the smallest number in the square.
To work out
the difference of the products of the opposite corners you have to
multiply the opposite numbers in the opposite corners by each other
then subtract. I will write these numbers in terms of n. I am using
a 2 x 2 square in a 10 x 10 grid.
= [(n + 1) x
(n + 10)] – [n( n + 11)]
= (n + 1n +
10n +10) – (n + 11n)
= (n + 11n +
10) – ( n + 11n)
= 10
Now I am
going to investigate what will happen if I change the grid sixe.
First I will
try a 5 x 5 grid.
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First I will
try a 2 x 2 square.
1
x 7
= 7 2
x 6
= 12 17 x 23 = 391 18 x 22 = 396
12 –
7 = 5 396
– 391 = 5
9 x 15 =
135 10 x 14 = 140
140 –
135 = 5
The
difference for a 2 x 2 square in a 5 x 5 grid always seems to be 5.
Now I will
try a 3 x 3 square.
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1
x 13
= 13 3
x 11
= 33
33 –
13 = 20
7
x 19
= 133 9
x 17
= 153
153 –
133 = 20
The
difference always seems to be 20 for a 3 x 3 square in a 5 x 5 grid.
Now I will
try a 4 x 4 square.
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1
x 19
= 19 4
x 16
= 64
64 –
19 = 45
2
x 20
= 40 5
x 17
= 85
85 –
40 = 45
The
difference is always 45.
I am going to
compare the results of the 10 x 10 grid to the results form the 5 x 5
grid.
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Square size
|
Grid size
|
|
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5 x 5
|
10 x 10
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2 x 2
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5
|
10
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3 x 3
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20
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40
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4 x 4
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45
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90
|
I can see
from this table that the differences of the products of the opposite
corners from the 5 x 5 grid are half of the differences from the 10 x
10 grid.
Eg. 10
2 = 5
- 2 = 20
- 2 = 45
This is
because the 5 x 5 grid is half the size of the 10 x 10. when the grid
size changes the position of the numbers change.
Now I am
going to try a different size grid, first I will try a 6 x 6 grid,
then a 7 x 7 grid.
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36
|
First I
will try a 2 x 2 square.
1
x 8
= 8 2
x 7
= 14 29
x 36
= 1044 35
x 30
= 1050
14 –
8 = 6
1050 – 1044 = 6
The
difference is always 6 in a 2 x 2 square in a 6 x 6 grid.
Now I will
try a 3 x 3 square.
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20
x 34
= 680 22
x 32
= 704
704 –
680 = 24
10
x 24
= 240 12
x 22
= 264
264 –
240 = 24
The
difference of the product of the opposite corners of a 3 x 3 square
in a 6x 6 grid is always 24.
Now I will
try a 4 x 4 square.
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36
|
3
x 24
= 72 6
x 21
= 126
126 –
72 = 54
13
x 34
= 442 16
x 31
= 496
496 –
442 = 52
The
difference is always 52.
Now I am
going to change the grid size to 7 x 7. I’ll try the 2 x 2 square
first.
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1
x 9
= 9 2
x 8
= 16 25
x 33
= 825 26
x 32
= 832
16 –
9 = 7
832 – 825 = 7
36 x 44 =
1584 43 x 37 = 1591
1591 –
1584 = 7
3 x 3
square.
15
x 31
= 465 17
x 29
= 493
493 465
= 28
4
x 20
= 80 6
x 18
= 108
108 –
80 = 28
The
difference is always 28 in a 3 x 3 square in a 7 x 7 grid.
4 x 4 square.
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4
x 28
= 112 7
x 25
= 175
175 –
112 = 63
22 x 46 =
1012 25 x 43 = 1075
1075 –
1012 = 63
The
difference is always 63 in a 4 x 4 square in a 7x 7 grid.
Now I
have tried some different sized grids I am going to put the results
into a table.
|
Square length (S)
|
Grid size (G)
|
|||
|
5 x 5
|
6 x 6
|
7 x 7
|
10 x 10
|
|
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2 x 2
|
5
|
6
|
7
|
10
|
|
3 x 3
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20
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24
|
28
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40
|
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4 x 4
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45
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54
|
63
|
90
|
I think
the formula to work out the difference of the products of the
opposite corners for any square in any sized grid, is connect I found
for any sized square in a 10 x 10 grid; 10(n – 1).
I have worked
out that the formula is (n – 1) x Grid size.
I will check
the formula by putting in some numbers.
Example
Grid size = 5
x 5
Square length
= 2
(2 – 1) x
5 From the table you can see that the difference of
1 x 5 =
5 the
products for the opposite corners in a 2 x 2
square in
a 10 x 10 grid is 5.
Proof
To prove the
formula works these numbers of the example with algebra.
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1
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2
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3
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4
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5
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6
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7
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9
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10
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17
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18
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19
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20
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21
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22
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23
|
24
|
25
|
I will
use n to represent the original number in top left of the square
inside the grid. I have highlighted it red
in the grid above. It is always the smallest number in the square.
To work out
the difference of the products of the opposite corners you have to
multiply the opposite numbers in the opposite corners by each other
then subtract. I will write these numbers in terms of n. I am using
a 2 x 2 square in a 5 x 5 grid.
= [(n +
1) x (n + 5)] – [n( n + 6)]
= (n + 1n +
5n +5) – (n + 6n)
= (n + 6n +
5) – ( n + 6n)
= 5
This is the
same answer that I got in the example above, and this is also the
same answer that is in the grid.
To further
the investigation I am investigate the difference of the products of
the opposite corners of rectangles inside a square grid.
First I will
investigate different sized rectangles inside a 10 x 10 grid.
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100
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2 x 3
rectangle
1
x 13
= 13 3
x 11
= 33 58
x 70
= 4060 60
x 68
=4080
33 –
13 = 20 4080
– 4060 = 20
5
x 26
= 130 6
x 25
= 150
150 –
130 = 20
The
difference of the products of the opposite corners is always 20 in a
2 x 3 rectangle in a 10 x 10 grid.
2 x 4
rectangle
7
x 38
= 266 8
x 37
= 296 55
x 86
= 4730 56
x 85
= 4760
296 –
266 = 30 4760
– 4730 = 30
31
x 44
= 1364 34
x 41
= 1394
1394 –
1394 =30
The
difference of the products of the opposite corners is always 30 for a
2 x 4 rectangle in a 10 x 10 grid.
2 x 5
rectangle
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41
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48
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56
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57
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60
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61
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64
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65
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66
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67
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68
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71
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76
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77
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78
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80
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81
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84
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85
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86
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87
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88
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90
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91
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92
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93
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95
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96
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97
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98
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99
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100
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1
x 15
= 15 5
x 11
= 55 76
x 90
=6840
80 x 86
= 6880
55 –
15 = 40
6880 – 6840 = 40
34
x 75
= 2550 35
x 74
= 2590
2590 –
2550 = 40
The
difference of the products of the opposite corners is always 40 in a
2 x 5 rectangle in a 10 x 10 grid.
3 x 4
rectangle
31
x 63
= 1953 33
x 61
= 2031 47
x 70
= 3290 50
x 67
= 3350
2031 –
1953 = 60 3350
– 3290 = 60
7
x 30
= 210 10
x 27
= 270
270 –
210 = 60
The
difference of the products of the opposite corners is always 60 in a
3 x 4 rectangle in a 10 x 10 grid.
4 x 5
rectangle
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40
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57
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59
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60
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61
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62
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63
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64
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65
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66
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67
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68
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69
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70
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71
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72
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73
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75
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76
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77
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78
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79
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80
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81
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82
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84
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86
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87
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88
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89
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90
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91
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92
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93
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94
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95
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96
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97
|
98
|
99
|
100
|
6
x 40
= 240 10
x 36
= 360 57
x 100
= 5700 60
x 97
= 5820
360 –
240 = 120
5820 - 5700 = 120
61
x 95
= 5795 65
x 91 = 5915
5915 –
5795 = 120
The
difference of the products of the opposite corners is always 120 in a
5 x 4 rectangle in a 10 x 10 grid.
5 x 6
rectangle
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99
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100
|
1
x 46
= 46 6
x 41
= 246
246 –
46 = 200
46
x 100
= 4600 50
x 96
= 4800
4800 –
4600 =200
The
difference of the products of the opposite corners is always 200 in a
5 x 6 rectangle in a 10 x 10 grid.
I will put my
results into a table.
|
Size of rectangle
|
Difference
|
|
2 x 3
|
20
|
|
2 x 4
|
30
|
|
2 x 5
|
40
|
|
3 x 4
|
60
|
|
4 x 5
|
120
|
|
5 x 6
|
200
|
I have
noticed that if you subtract 1 from each length of the rectangle then
multiply them both by 10 then you get the difference.
X= 1st
length of the rectangle
Y= 2nd
length of the rectangle
The formula
is: 10(x – 1)(y – 1)
I will
fit some numbers into the formula.
2 x 3 2 –
1 = 1 This is the same answer that I found
3 – 1 =
2 in my results.
1 x 2
x 10 = 20
Proof
Again I will
I use algebra to prove that this formula works.
N= the
number at the top left corner of the rectangle (smallest number)
I am using a
2 x 3 rectangle in the 10 x 10 grid.
= [(n + 2) x
(n + 10)] – [n(n +12)
=n + 2n + 10n
+ 20 – n + 12n
=n + 12n + 20
– n +12n
= 20
The answer is
20 as it was in the example above and the results I collected earlier
for a 2 x 3 rectangle.
Now I
am going to investigate changing the grid size.
First I will
try a 5 x 5 grid.
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17
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18
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19
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20
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21
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22
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23
|
24
|
25
|
2 x 3
rectangle
1
x 8
= 8 3
x 6
= 18
18 –
8 = 10
14
x 25
= 350 15
x 24
= 360
360 –
350 = 10
The
difference of the products of the opposite corners is always 10 for a
2 x 3 rectangle in a 5 x 5 grid.
2 x 4
rectangle
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2
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3
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4
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5
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20
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21
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22
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23
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24
|
25
|
1
x 9
= 9 4
x 6
= 24 7 x 23 =
161 8 x 22 = 176
24 –
9 = 15 176
– 161 = 15
The
difference of the products of the opposite corners is always 15 for a
2 x 4 rectangle in a 5 x 5 grid.
|
1
|
2
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3
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4
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5
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6
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7
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8
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9
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10
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12
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21
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22
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23
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24
|
25
|
1
x 10
= 10 5
x 6
= 30 4 x 25 = 100 5 x 24 =120
30 –
10 = 20 120
– 100 = 20
The
difference of the products of the opposite corners is always 20 for a
2 x 5 rectangle in a 5 x 5 grid.
3 x 4
rectangle
|
1
|
2
|
3
|
4
|
5
|
|
6
|
7
|
8
|
9
|
10
|
|
11
|
12
|
13
|
14
|
15
|
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16
|
17
|
18
|
19
|
20
|
|
21
|
22
|
23
|
24
|
25
|
1
x 14
= 14 4
x 11
= 44 8
x 25
= 200 10
x 23
= 230
44 –
14 = 30 230
– 200 = 30
|
Rectangle size
|
Grid size
|
|
|
5 x 5
|
10 x 10
|
|
|
2 x 3
|
10
|
20
|
|
2 x 4
|
15
|
30
|
|
2 x 5
|
20
|
40
|
|
3 x 4
|
30
|
60
|
From
this table I can see that the differences are half in the 5 x 5 then
in the 10 x 10 grid.
Now I will
try 6 x 6 grid.
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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12
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13
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14
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15
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16
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17
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18
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19
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20
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21
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22
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23
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24
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25
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26
|
27
|
28
|
29
|
30
|
|
31
|
32
|
33
|
34
|
35
|
36
|
2 x 3
rectangle
1
x 9
= 9 3
x 7
= 21 19
x 32
= 608 20
x 31
= 620
21 –
9 = 12 620
– 608 = 12
The
difference is always 12 in a 2 x 3 rectangle in a 6 x 6 grid.
2 x 4
rectangle
5 x 24
= 120 6 x 23 = 138 27 x 36 = 972 30 x 33 = 990
138 –
120 = 18
990 – 972 = 18
The
difference is always 18 in a 2 x 4 rectangle in a 6 x 6 grid.
2 x 5
rectangle
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1
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2
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3
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4
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5
|
6
|
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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18
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19
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20
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21
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22
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23
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24
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25
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26
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27
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28
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29
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30
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31
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32
|
33
|
34
|
35
|
36
|
1
x 11
= 11 5
x 7
= 35 20
x 30
= 600 24
x 26
= 624
35 –
11 = 24 624
– 600 = 24
The
difference is always 24 in a 2 x 5 rectangle in a 6 x 6 grid.
3 x 4
rectangle
|
1
|
2
|
3
|
4
|
5
|
6
|
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7
|
8
|
9
|
10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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21
|
22
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23
|
24
|
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25
|
26
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27
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28
|
29
|
30
|
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31
|
32
|
33
|
34
|
35
|
36
|
1 x 16 =
16 4 x 13 = 52 15 x 35 = 525 17 x 33 = 561
52 –
16 = 36 561
– 525 = 36
The
difference is always 36 in a 3 x 4 rectangle in a 6 x 6 grid.
Now I am
going to change the grid size to 7 x 7.
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1
|
2
|
3
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4
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5
|
6
|
7
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8
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9
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10
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14
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15
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17
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18
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19
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20
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21
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22
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23
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24
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25
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26
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27
|
28
|
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29
|
30
|
31
|
32
|
33
|
34
|
35
|
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36
|
37
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38
|
39
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40
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41
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42
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43
|
44
|
45
|
46
|
47
|
48
|
49
|
2 x 3
rectangle
1
x 10
= 10 3
x 8
= 24 33
x 48
= 1584 34
x 47
= 1598
24 –
10 = 14 1598
– 1584 = 14
The
difference is always 14 in a 2 x 3 rectangle in a 7 x 7 grid.
2 x 4
rectangle
4
x 14
= 56 7
x 11
= 77 22
x 32
= 704 25
x 29
= 725
77 –
56 = 21 725
– 704 = 21
The
difference is always 21 in a 2 x 4 rectangle in a 7 x 7 grid.
2 x 5
rectangle
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
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15
|
16
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17
|
18
|
19
|
20
|
21
|
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22
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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
|
1
x 12
= 12 5
x 8
= 40 20
x 49
= 980 21
x 48
= 1008
40 –
12 = 28 1008
– 980 = 28
The
difference is always 28 in a 2 x 5 rectangle in a 7 x 7 grid.
3 x 4
rectangle
15
x 32
= 480 18
x 29
= 522 23
x 47
= 1081 23
x 44
= 1144
522 –
480 = 42 1144
– 1081 = 42
The
difference is always 28 in a 3 x 4 rectangle in a 7 x 7 grid.
I will put my
results into a table.
|
Size of rectangle
|
Size of grid
|
|||
|
5 x 5
|
6 x 6
|
7 x 7
|
10 x 10
|
|
|
2 x 3
|
10
|
12
|
14
|
20
|
|
2 x 4
|
15
|
18
|
21
|
30
|
|
2 x 5
|
20
|
24
|
28
|
40
|
|
3 x 4
|
30
|
36
|
42
|
60
|
I think that
the formula to work out the difference of the products of the
opposite corners of any rectangle in any sized square grid is similar
to the formula I found earlier for the difference of any rectangle in
a 10 x 10 grid, 10(x – 1)(y - 1).
The formula
is almost the same except it is multiplied by the grid size not 10.
G=Grid
size
Formula: G(x
– 1)(y – 1)
This is
connected to the formula which can work the difference of the
products of the opposite corners of any sized square in any sized
square grid. That formula is: G(x – 1)
This can also
be written as: G(x – 1)(x – 1)
From this you
can see that the two formula are connected. The only way they are
different is that in the rectangle there is x and y because the two
lengths of the rectangle are different, and the square formula only
has x as both lengths are the same.
Proof
I will use
algebra again to show that the formula G(x – 1)(y – 1) works.
N= the number
at the top left corner of the rectangle (smallest number).
I will use a
3 x 4 rectangle in a 5 x 5 square.
=[(n + 2)(n +
5)] – [n(n + 7)
= n + 7n + 10
– n + 7n
= 10
From my table
I can see that this is the same answer I got before.
Thank you very much about sharing all details of square rectangle products of opposites. I have been always confused about this topic but have now good knowhow.
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