GCSE Foundation and Intermediate Level
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Simon Langley 9RL
I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a one hundred square. For example:
54
55
56
54 and 66
64 and 56
64
65
66
54 x 66 = 3564
"
3584 - 3564 = 20
64 x 56 = 3584
The difference of this rectangle is 20.
Firstly I am going to look at rectangles of size 2 x 2 as these are the smallest possible rectangles with four corners...
2
"
x 12 = 12
22 - 12 = 10
1
2
1 x 2 = 22
4
5
"
4 x 15 = 60
70 - 60 = 10
4
5
4 x 5 = 70
88
89
"
88 x 99 = 8712
8722 - 8712 = 10
98
99
98 x 89 = 8722
From these results I can conclude that every 2 x 2 rectangle has a difference of 10. I am now going to try rectangles of size 2 x m (where m is the other side).
2 x 3:
3
"
33 - 11 = 20
1
3
4
4
"
84 - 64 = 20
4
6
7
9
"
53 - 133 = 20
7
9
This shows that all 2 x 3 rectangles have a difference of 20
2 x 4:
4
"
44 - 14 = 30
1
4
5
8
"
20 - 90 = 30
5
8
21
24
"
744 - 714 = 30
31
34
This shows that all 2 x 4 rectangles have a difference of 30
2 x 5:
"
55 - 15 = 40
1
5
6
0
"
60 - 120 = 40
2
3
4
5
6
7
8
9
0
1
2
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9
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
O
P
P
O
S
I
T
E
50
51
C
O
R
N
E
R
S
59
60
61
62
63
64
65
66
67
68
69
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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
Simon Langley 9RL
I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a one hundred square. For example:
54
55
56
54 and 66
64 and 56
64
65
66
54 x 66 = 3564
"
3584 - 3564 = 20
64 x 56 = 3584
The difference of this rectangle is 20.
Firstly I am going to look at rectangles of size 2 x 2 as these are the smallest possible rectangles with four corners...
2
"
x 12 = 12
22 - 12 = 10
1
2
1 x 2 = 22
4
5
"
4 x 15 = 60
70 - 60 = 10
4
5
4 x 5 = 70
88
89
"
88 x 99 = 8712
8722 - 8712 = 10
98
99
98 x 89 = 8722
From these results I can conclude that every 2 x 2 rectangle has a difference of 10. I am now going to try rectangles of size 2 x m (where m is the other side).
2 x 3:
3
"
33 - 11 = 20
1
3
4
4
"
84 - 64 = 20
4
6
7
9
"
53 - 133 = 20
7
9
This shows that all 2 x 3 rectangles have a difference of 20
2 x 4:
4
"
44 - 14 = 30
1
4
5
8
"
20 - 90 = 30
5
8
21
24
"
744 - 714 = 30
31
34
This shows that all 2 x 4 rectangles have a difference of 30
2 x 5:
"
55 - 15 = 40
1
5
6
0
"
60 - 120 = 40