In this assignment I am going to investigate about the number grid. I am going to use two different types of formula to find out the correct answers one is numerical formula and second is algebra.
In numerical way I am going to multiply the top left number with right bottom number and the top right number with bottom left number and than I am going to find the difference between the products. The difference of product is going to be same as the size of a grid (in width) in 2 x 2 size. In 3 x 3 size the difference going to be four times bigger than a grid (width) size. I will also use the 5 x 5 grid, 6 x 6 grid and 7 x 7 grid to see how it works.
E.g.
5 x 5 Grid
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
20 x 24 = 480
9 x 25 = 475
Difference between these products is 5.
In algebra I am going to use different formula to find out the correct answer.
This is the formula that I am going to use:
n = Top left corner
n + 1 = Top right corner
n + 5 = Bottom left corner
n + 6 = Bottom right corner
(n + 1) (n + 5) - (n) (n + 6)
n2 + 5n + n + 5 - n2 - 6n (n2 cancel out with - n2, 5n + n cancel out with - 6n)
5 left over
This is the product difference of 2 x 2 in a 5 x 5 grid.
8 x 8 Grid
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
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
1 x 18 = 198
0 x 19 = 190
Difference between these products is 8.
n = Top left corner
n + 1 = Top right corner
n + 8 = Bottom left corner
n + 9 = Bottom right corner
(n + 1) (n + 8) - (n) (n + 9)
n2 + 8n + n + 8 - n2 - 9n (n2 cancel out with - n2, 8n + n cancel out with - 9n)
8 left over
This is the product difference of 2 x 2 in an 8 x 8 grid.
9 x 9 Grid
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
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
26 x 34 = 884
25 x 35 = 875
Difference between these products is 9.
n = Top left corner
n + 1 = Top right corner
n + 9 = Bottom left corner
n + 10 = Bottom right corner
(n + 1) (n + 9) - (n) (n + 10)
n2 + 9n + n + 9 - n2 - 10n (n2 cancel out with - n2, 9n + n cancel out with - 10n)
9 left over
This is the product difference of 2 x 2 in a 9 x 9 grid.
0 x 10 Grid
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
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
00
48 x 57 = 2736
47 x 58 = 2726
Difference between these products is 10.
n = Top left corner
n + 1 = Top right corner
n + 10 = Bottom left corner
n + 11 = Bottom right corner
(n + 1) (n + 10) - (n) (n + 11)
n2 + 10n + n + 10 - n2 - 11n (n2 cancel out with - n2, 10n + n cancel out with - 11n)
0 left over
This is the product difference of 2 x 2 in a 10 x 10 grid.
1 x 11 Grid
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
In numerical way I am going to multiply the top left number with right bottom number and the top right number with bottom left number and than I am going to find the difference between the products. The difference of product is going to be same as the size of a grid (in width) in 2 x 2 size. In 3 x 3 size the difference going to be four times bigger than a grid (width) size. I will also use the 5 x 5 grid, 6 x 6 grid and 7 x 7 grid to see how it works.
E.g.
5 x 5 Grid
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
20 x 24 = 480
9 x 25 = 475
Difference between these products is 5.
In algebra I am going to use different formula to find out the correct answer.
This is the formula that I am going to use:
n = Top left corner
n + 1 = Top right corner
n + 5 = Bottom left corner
n + 6 = Bottom right corner
(n + 1) (n + 5) - (n) (n + 6)
n2 + 5n + n + 5 - n2 - 6n (n2 cancel out with - n2, 5n + n cancel out with - 6n)
5 left over
This is the product difference of 2 x 2 in a 5 x 5 grid.
8 x 8 Grid
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
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
1 x 18 = 198
0 x 19 = 190
Difference between these products is 8.
n = Top left corner
n + 1 = Top right corner
n + 8 = Bottom left corner
n + 9 = Bottom right corner
(n + 1) (n + 8) - (n) (n + 9)
n2 + 8n + n + 8 - n2 - 9n (n2 cancel out with - n2, 8n + n cancel out with - 9n)
8 left over
This is the product difference of 2 x 2 in an 8 x 8 grid.
9 x 9 Grid
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
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
26 x 34 = 884
25 x 35 = 875
Difference between these products is 9.
n = Top left corner
n + 1 = Top right corner
n + 9 = Bottom left corner
n + 10 = Bottom right corner
(n + 1) (n + 9) - (n) (n + 10)
n2 + 9n + n + 9 - n2 - 10n (n2 cancel out with - n2, 9n + n cancel out with - 10n)
9 left over
This is the product difference of 2 x 2 in a 9 x 9 grid.
0 x 10 Grid
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
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
00
48 x 57 = 2736
47 x 58 = 2726
Difference between these products is 10.
n = Top left corner
n + 1 = Top right corner
n + 10 = Bottom left corner
n + 11 = Bottom right corner
(n + 1) (n + 10) - (n) (n + 11)
n2 + 10n + n + 10 - n2 - 11n (n2 cancel out with - n2, 10n + n cancel out with - 11n)
0 left over
This is the product difference of 2 x 2 in a 10 x 10 grid.
1 x 11 Grid
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
