Shaun Lalley 10 Bingham
Maths Coursework: Number Grids
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
2 x 2 Grids
I will start by investigating what happens if we take a 2x2 grid from a 10x10 master grid. I will multiply the top right number by the lower left. Then I will subtract the lower right number multiplied by the top left e.g.
a
b
c
d
axd=ab
cxb=cd ab-cd
2
1
2
x 12 =12 22 - 12=10
2 x 11 =22
83
84
93
94
84 x 93 =7812 7812 - 7802=10
83 x 94 =7802
89
90
99
00
90 x 99 =8910 8910 - 8900=10
89 x 100 =8900
9
0
9
20
0 x 19 =190 190 - 180=10
9 x 20 =180
The answer is always 10. I predict that my next 2 x 2 grid will result in the answer 10.
6
7
6
7
6 x 17 =102 102 - 92=10
7 x 16 =92
My prediction was correct. I will now prove that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10, using algebra.
X
X+1
X+10
X+11
(X+1)(X+10) = X2 + 11X + 10
X(X + 11) =X2 + 11X
(X2 + 11X + 10) - (X2 + 11X) = 10
This proves that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10.
3 x 3 Grids
I will now investigate 3x3 grids
5
6
7
5
6
7
25
26
27
7 x 25 = 175
5 x 27 = 135
75 - 135 = 40
2
3
4
22
23
24
32
33
34
4 x 32 = 448
2 x 34 = 408
448 - 408 = 40
48
49
50
58
59
60
68
69
70
50 x 68 = 3400
48 x 70 = 3360
3400 - 3360 = 40
71
72
73
81
82
83
91
92
93
73 x 91 = 6643
71 x 93 = 6603 6643 - 6603 = 40
I predict that my next 3 x 3 grid will result in the answer 40.
43
44
45
53
54
55
63
64
65
43 x 65 = 2795
45 x 63 = 2835
2835 - 2795 = 40
My prediction was correct. I will now prove that all 3 x 3 grids taken from a 10 x 10 master grid result in an answer of 40, using algebra.
X
X+1
X+2
X+10
X+11
X+12
X+20
X+21
X+22
(X+2)(X+20) = X2 + 22X + 2X + 40
X(X+22) = X2 + 22X
(X2 + 22X + 40) - (X2 + 22X) = 40
This proves hat all 3 x 3 grids taken from a 10 x 10 master grid result in the answer 40.
4 x 4 Grids
I will now investigate 4x4 grids
2
3
4
1
2
3
4
21
22
23
24
31
32
33
34
4 x 31 = 124
x 34 = 34
24 - 34 = 90
44
45
46
47
54
55
56
57
64
65
66
67
74
75
76
77
47 x 74 = 3478
44 x 77 = 3388
3478 - 3388 = 90
5
6
7
8
5
6
7
8
25
26
27
28
35
36
37
38
8 x 35 = 280
5 x 38 = 190
280 - 190 = 90
I predict that my next 4 x 4 grid will result in the answer 90.
67
68
69
70
77
78
79
80
87
88
89
90
97
98
99
00
70 x 97 = 6790
67 x 100 = 6700
6790 - 6700 = 90
I will now use algebra to prove that all 4 x 4 grids taken from a 10 x 10 master grid result in an answer of 90.
Maths Coursework: Number Grids
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
2 x 2 Grids
I will start by investigating what happens if we take a 2x2 grid from a 10x10 master grid. I will multiply the top right number by the lower left. Then I will subtract the lower right number multiplied by the top left e.g.
a
b
c
d
axd=ab
cxb=cd ab-cd
2
1
2
x 12 =12 22 - 12=10
2 x 11 =22
83
84
93
94
84 x 93 =7812 7812 - 7802=10
83 x 94 =7802
89
90
99
00
90 x 99 =8910 8910 - 8900=10
89 x 100 =8900
9
0
9
20
0 x 19 =190 190 - 180=10
9 x 20 =180
The answer is always 10. I predict that my next 2 x 2 grid will result in the answer 10.
6
7
6
7
6 x 17 =102 102 - 92=10
7 x 16 =92
My prediction was correct. I will now prove that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10, using algebra.
X
X+1
X+10
X+11
(X+1)(X+10) = X2 + 11X + 10
X(X + 11) =X2 + 11X
(X2 + 11X + 10) - (X2 + 11X) = 10
This proves that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10.
3 x 3 Grids
I will now investigate 3x3 grids
5
6
7
5
6
7
25
26
27
7 x 25 = 175
5 x 27 = 135
75 - 135 = 40
2
3
4
22
23
24
32
33
34
4 x 32 = 448
2 x 34 = 408
448 - 408 = 40
48
49
50
58
59
60
68
69
70
50 x 68 = 3400
48 x 70 = 3360
3400 - 3360 = 40
71
72
73
81
82
83
91
92
93
73 x 91 = 6643
71 x 93 = 6603 6643 - 6603 = 40
I predict that my next 3 x 3 grid will result in the answer 40.
43
44
45
53
54
55
63
64
65
43 x 65 = 2795
45 x 63 = 2835
2835 - 2795 = 40
My prediction was correct. I will now prove that all 3 x 3 grids taken from a 10 x 10 master grid result in an answer of 40, using algebra.
X
X+1
X+2
X+10
X+11
X+12
X+20
X+21
X+22
(X+2)(X+20) = X2 + 22X + 2X + 40
X(X+22) = X2 + 22X
(X2 + 22X + 40) - (X2 + 22X) = 40
This proves hat all 3 x 3 grids taken from a 10 x 10 master grid result in the answer 40.
4 x 4 Grids
I will now investigate 4x4 grids
2
3
4
1
2
3
4
21
22
23
24
31
32
33
34
4 x 31 = 124
x 34 = 34
24 - 34 = 90
44
45
46
47
54
55
56
57
64
65
66
67
74
75
76
77
47 x 74 = 3478
44 x 77 = 3388
3478 - 3388 = 90
5
6
7
8
5
6
7
8
25
26
27
28
35
36
37
38
8 x 35 = 280
5 x 38 = 190
280 - 190 = 90
I predict that my next 4 x 4 grid will result in the answer 90.
67
68
69
70
77
78
79
80
87
88
89
90
97
98
99
00
70 x 97 = 6790
67 x 100 = 6700
6790 - 6700 = 90
I will now use algebra to prove that all 4 x 4 grids taken from a 10 x 10 master grid result in an answer of 90.
