This investigation is about finding the difference between the products of the opposite corner numbers in a number square. There are three variables which I can change whilst doing my investigation, they are the size of the grid, the shape of the grid and the numbers inside the grid. I am going to start by looking only at number squares with consecutive numbers
Consecutive Numbers
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
4 x 13 = 52
1 x 16 = 16
Difference = 36
0
3
12
15
0 x 15 = 0
3 x 12 = 36
Difference = 36
-2
1
10
13
1 x 10 = 10
-2 x 13 = -26
Difference = 36
The difference seems to be the same, for these 3 the answer is 36 but this isn’t proof.
Let X stand for the start number which can be any real number.
X
X + 3
X + 12
X + 15
(X + 3) (X + 12) = X2 + 3X + 12X + 36
= X2 + 15X + 36
X (X + 15) = X2 + 15X
Difference = 36
So, the difference between the products of the opposite corner numbers in a 4×4 number square is 36. What about a 3×3 number square?
X
X + 2
X + 6
X + 8
(X + 2) (X + 6) = X2 + 2X + 6X +12
= X2 + 8X +12
X (X + 8) = X2 + 8X
Difference = 12
So, the difference between the products of the opposite corner numbers in a 3×3 number square is 10.
What about Other squares?
X
This investigation does not work with a square size of 1×1, as the square does not have four corners.
X
X + 1
X + 2
X + 3
(X + 1) (X + 2) = X2 + X + 2X +2
= X2 + 3X +2
X (X + 3) = X2 + 3X
2
X
X + 4
X + 20
X + 24
(X + 4) (X + 20) = X2 + 4X + 20X + 80
= X2 + 24X + 80
X (X + 24) = X2 + 24X
= 80
X
X + 5
X + 30
X + 35
(X + 5) (X + 30) = X2 + 5X + 30X + 150
= X2 + 35X + 150
X (X + 35) = X2 + 35X
150
Square size
Difference
Factors
2×2
2
2×1
3×3
12
3×4
4×4
36
4×9
5×5
80
5×16
6×6
150
6×25
10×10
?
?
NxN
N(N – 1)2
Predict + check
Looking at the patterns of numbers from my tables of results it appears for a grid size of NxN the difference is N(N – 1)2.
I predict that for a 10×10 grid the difference will be 10 x 92 = 10 x 81 = 810.
I will check by drawing.
X
X+9
X+90
X+99
(X + 9) (X + 90) = X2 + 9X + 90X + 810
= X2 + 99X + 810
X(X + 99) = X2 + 99X
810
The check shows that the predicted formula is correct. But this is not proof.
X
X+(N-1)
X+N(N-1)
X+N(N-1)+(N-1)
(X + (N-1)) (X + N(N-1)) = X2 + X(N-1) + X(N(N-1)) + N(N-1)(N-1)
X (X + N(N-1) + N-1)) = X2 + X(N-1) + X(N(N-1))
N(N-1)(N-1)
=N(N-1)2
This formula is the same as before so I have proved my prediction.
Grid within a grid
The formula that I have figured out works for any sized square with a consecutive number grid but what about a grid within a grid?
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83
84
85
86
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89
90
91
92
93
94
95
96
97
98
99
100
I’m now going to see whether the corners have any algebraic relation to each other.
35
36
45
46
66
68
86
88
17
20
47
50
The algebraic terms for the corners seems to be the same for any outer square so I’ll now put these terms into the square and find the difference in algebraic terms.
X
X+(R-1)
X+P(R-1)
X+P(R-1)+(R-1)
(X + (R-1)) (X + P(R-1)) = X2 + XP (R-1) + X(R-1) + P(R-1)(R-1)
= X2 + XP(R – 1) + X(R – 1) + P(R-1)2
X (X + P(R-1) + (R -1 ) = X2 + XP(R – 1) + X(R – 1)
= P(R-1)2
Predict + check
Looking at the results I believe that inside a square PxP the difference of the products of opposite numbers in a inner square sized RxR = P (R-1)2. I predict that, for a 6×6 square inside a 10×10 square that the difference will be 10 x (6-1)2 = 10 x 25 = 250.
I will check by drawing.
14
19
64
69
19 x 64 = 1216
14 x 69 = 966
= 250
My equation is right.
I Have noticed that the height of the outer square is irrelevant in the formula so this formula will also work for squares inside rectangles.
Rectangles
I have worked out the formula in number squares, but what about number rectangles?
1
2
3
4
5
6
7
8
9
10
11
12
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19
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25
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30
31
32
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35
36
37
38
39
40
10 x 31 = 310
1 x 40 = 40
= 270
I’m now going to see whether the corners have any algebraic relation to each other.
1
10
21
30
1
10
41
50
X
X+(N-1)
X+(M-1)N
X+(N-1) + (M-1)N
(X + (N-1)) ((X + (M-1)N) =X2 + XN(M-1) + X(N-1) + N(N-1)(M-1)
X(X + (N – 1) + (M-1)N)= X2+ XN(M-1) + X(N-1)
Difference = N(N-1)(M-1)
Check
Using my equation I predict that for a rectangles sized 7×5 the difference will be
N(N-1)(M-1) = 7(7-1)(5-1) = 7 X 6 X 4 = 168.
1
2
3
4
5
6
7
8
9
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19
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25
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29
30
31
32
33
34
35
7 X 29 = 203
1 X 35 = 35
Difference = 168
My equation is correct
Rectangles inside rectangles
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120
47
50
62
65
67
72
97
102
91
95
106
110
26
28
41
43
X
X+(C-1)
X+A(D-1)
X+(C-1)+A(D-1)
(X + (C-1)) (X + A(D-1)) = X2 + XA(D-1) + X(C-1) + A(D-1)(C-1)
(X) (X + (C-1) + A(D-1)) = X2 + XA(D-1) + X(C-1)
Difference = A(D-1)(C-1)
Check
Using my equation, I predict that inside a 6×5 rectangle a 3×2 inner rectangle will have a difference of 6(2-1)(3-1) = 6 x 1 x 2 = 12.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8
10
14
16
10 x 14 = 140
8 x 16 = 128
Difference = 12
My equation is correct.
Looking at these results I have realised that squares are actually rectangles where the length = width so I am now not going to treat squares and rectangles differently.
Patterns
What happens when I change the pattern inside a rectangle?
Arithmetic Progressions
2
8
18
24
2
6
14
18
3
12
39
48
To get the answer to the equation, you have to multiply the rectangle going up in consecutive numbers by the arithmetic number you are using this is because when you use consecutive numbers, you are actually going up in arithmetic progression of 1.
X
X+S(N-1)
X+SN(N-1)
X+(SN(N-1)+S(N-1))
(X + S(N-1)) (X + SN(M-1) = X2 + XSN(M-1) + XS(N-1) + SN(M-1) S(N-1)
X (X + (SN(M-1) + S(N-1) = X2 + XSN(M-1) + XS(N-1)
Difference = SN(M-1) S(N-1)
Check
For a table 6×6 with an arithmetic progression of nine I predict that the difference will be (9X6(6-1)) X (9(6-1) = (54 X 5) X (9 X 5) = 270 X 45 = 12150
9
54
279
324
54 X 279 = 15066
9 X 324 = 2916
Difference = 12150
My equation was right.
Geometric Progressions
There is a pattern for arithmetic progressions but what about geometric progressions?
2
16
8192
65536
2
8
16
64
3
27
2187
19683
aX
aX+(N-1)
aX+N(N-1)
aX+N(N-1)+(N-1)
aX+N(N-1) aX+(N-1)= X(XN) + XN (XN(N-1))
aX aX+N(N-1)+(N-1) = X(XN) + XN (XN(N-1))
Difference = 0
Check
For a 3×3 table with a geometric progression of 7 I predict that the difference is 0.
7
343
823543
40353607
343 X 823543 = 282475249
7 X 40353607 = 282475249
Difference = 0
My equation is right.
Arithmetic Progressions in grids within grids
I have looked at grids inside a grid and I have also looked at arithmetic progressions but what happens when I put the together?
2
4
6
8
10
12
14
16
18
20
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24
26
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30
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34
36
38
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128
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168
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200
136
138
156
158
24
28
44
48
88
92
128
132
12
20
52
60
X
X + S(C-1)
X + SA(D-1)
X + S(C-1) + SA(D-1)
(X + S(C-1)) (X + SA(D-1)) = X2 + SAX(D-1) + SX(C-1) + SA(D-1)S(C-1)
(X) (X + S(C-1) + SA(D-1)) = X2 + SAX(D-1) + SX(C-1)
Difference = SA(D-1)S(C-1)
Check
Using my formula I predict that for a 6×4 outer grid with an arithmetic progression of 5 the difference of the product of the opposite corners inside an inner grid of 3×2 will equal 5×6(2-1)5(3-1) = 30(1)5(2) = 30 x 10 = 300.
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
15
25
45
55
25 x 45 = 1125
15 x 55 = 825
Difference = 300
My equation is correct.
Geometric progressions in grids within grids
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25
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28
29
210
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288
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291
292
293
294
295
296
297
298
299
2100
After doing my original study on geometric progressions I have realised that it is easier to keep the numbers in powers form.
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NX
NX+(C-1)
NX+A(D-1)
NX+(C-1)+A(D-1)
NX + (C-1) X NX + A(D-1) = N2X + (C-1) + A(D-1)
NX X NX + (C-1) + A(D-1) = N2X + (C-1) + A(D-1)
Difference = 0
Check
For a 7×3 outer grid a 4×2 inner grid with a geometric progression of 7 will be 0.
71
72
73
74
75
76
77
78
79
710
711
712
713
714
715
716
717
718
719
720
721
73
74
75
76
710
711
712
713
76 + 710 = 716
73 + 713 = 716
Difference = 0
My equation is correct.
Spirals
What would happen if I spiralled into the centre?
1
2
3
4
5
6
20
21
22
23
24
7
19
32
33
34
25
8
18
31
36
35
26
9
17
30
29
28
27
10
16
15
14
13
12
11
1
9
19
11
8
10
14
12
X
X+(N-1)
X+2(N-1)+(M-1)
X+(N-1)+(M-1)
(X+(N-1)) (X+2(N-1)+(M-1) = X2+X(N-1)+ X(M-1)+2X(N-1) +2(N-1)2+(M-1)(N-1)
(X) (X+(N-1)+(M-1) = X2+X(N-1)+X(M-1)
Difference = 2X(N-1)+2(N-1)2+(M-1)(N-1)
Check
For a 7×5 rectangle with a starting number of 3 I predict that the difference will be
2×3(7-1)+2(7-1)2+(5-1)(7-1) = 6(6)+2(6)2+(4)(6) = 36+2(36)+24 = 36+72+24 = 132.
3
9
19
13
9 x 19 = 171
3 x 13 = 39
Difference = 132
My equation is correct.
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