Number Grids
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0
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This grid shows the numbers one to one hundred. The two of the corners diagonally opposite each other on a grid like this can be multiplied together to make a number. The other to corners can be multiplied to give a second number; there is a difference between the two figures. This difference is the interesting figure. We can compare the difference between grids of different sizes.
2
1
2
Let us start of small with a 2x2 grid.
We can work out the difference using the long method first.
x12 = 12
2x11 = 22
22-12 = 10. Therefore, ten is the difference of our first example. Quite easy you would think, but nerveless there are three sums, And, as the numbers get bigger the sums get harder.
N
N+1
N+10
N+11
So how can we get around this? We will use algebra.
This grid (right) represents a 2x2 grid. It can be used with the correct formulas to work out any difference for a 2x2 grid. So we start with a formula to find the two numbers multiplied together.
n² (n+11), but we can improve on this buy tuning the to formulas into 1.
In this second formula, we put both sums together.
(n+1) (n+10) when we expand the brackets, it reveals the difference for any 2x2 grid.
(n+1) (n+10) = n² + 11n + 10 so this method works and proves that the difference is 10.
However, will our formula continue to work as the sizes of grid become larger? So on to a 3x3 grid.
This time the numbers on our grid dose not start with one, but our formula is designed to work with any size of grid and any numbers, so we should be ok.
31
32
33
41
42
43
51
52
53
N
N+1
N+2
N+10
N+11
N+12
N+20
N+21
N+22
Let us try our formula first.
( n + 2 ) ( n + 20 ) =
n² + 20n + 40
So 40 is our difference according to the formula. However, lets check using the basic method, seen as it is only the second time it has been used.
31x53 = 1643
33x51 = 1683
683-1646 = 40
It is confirmed and our formula works once again.
7
8
9
0
7
8
9
20
27
28
29
30
37
38
39
40
Next, we will try a 4x4 square the principles stay the same, just with bigger numbers. So we have our grid (left).
Moreover, we have our formula grid, which can be used for working out the difference for any 4x4 grid. (Below)
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
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40
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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
00
This grid shows the numbers one to one hundred. The two of the corners diagonally opposite each other on a grid like this can be multiplied together to make a number. The other to corners can be multiplied to give a second number; there is a difference between the two figures. This difference is the interesting figure. We can compare the difference between grids of different sizes.
2
1
2
Let us start of small with a 2x2 grid.
We can work out the difference using the long method first.
x12 = 12
2x11 = 22
22-12 = 10. Therefore, ten is the difference of our first example. Quite easy you would think, but nerveless there are three sums, And, as the numbers get bigger the sums get harder.
N
N+1
N+10
N+11
So how can we get around this? We will use algebra.
This grid (right) represents a 2x2 grid. It can be used with the correct formulas to work out any difference for a 2x2 grid. So we start with a formula to find the two numbers multiplied together.
n² (n+11), but we can improve on this buy tuning the to formulas into 1.
In this second formula, we put both sums together.
(n+1) (n+10) when we expand the brackets, it reveals the difference for any 2x2 grid.
(n+1) (n+10) = n² + 11n + 10 so this method works and proves that the difference is 10.
However, will our formula continue to work as the sizes of grid become larger? So on to a 3x3 grid.
This time the numbers on our grid dose not start with one, but our formula is designed to work with any size of grid and any numbers, so we should be ok.
31
32
33
41
42
43
51
52
53
N
N+1
N+2
N+10
N+11
N+12
N+20
N+21
N+22
Let us try our formula first.
( n + 2 ) ( n + 20 ) =
n² + 20n + 40
So 40 is our difference according to the formula. However, lets check using the basic method, seen as it is only the second time it has been used.
31x53 = 1643
33x51 = 1683
683-1646 = 40
It is confirmed and our formula works once again.
7
8
9
0
7
8
9
20
27
28
29
30
37
38
39
40
Next, we will try a 4x4 square the principles stay the same, just with bigger numbers. So we have our grid (left).
Moreover, we have our formula grid, which can be used for working out the difference for any 4x4 grid. (Below)
