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Opposite Corners Investigation

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Opposite Corners Investigation

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

I am now only going to type the corner numbers, as the other numbers
are irrelevant




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 4x4 number square is 36. What about a 3x3 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 3x3 number square is 10. What about Other squares?

X

This investigation does not work with a square size of 1x1, 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

2x2

2

2x1

3x3

12

3x4

4x4

36

4x9

5x5

80

5x16

6x6

150

6x25

10x10

?

?

Text Box: N is the length of the square


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 10x10 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.

These are the equations that I have figured out for each corner.

[IMAGE]X

X+(N-1)

These are the equations that I have figured out for each corner.

These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]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?

1

2

3

4

5

6

7

8

9

10

P is the length of the outer grid.

R is the length of the inner grid.

X is the start number.

11

12

13

14

15

16

17

18

19

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

100

I'm now going to see whether the corners have any algebraic relation
to each other.

[IMAGE]

[IMAGE]




This number is the starting number + the top right corner + the bottom
left corner so I'll write this out in equation form

35

X + (R - 1)

[IMAGE]36

[IMAGE]

X + P (R-1)

[IMAGE]45

X + (R-1) + (P(R-1))

[IMAGE]46

[IMAGE]

45 is the starting number + 10 so I need to find a way to add 10



X

X + (R - 1)




68 is the starting number + 2 so I need to find a way to add 2.

[IMAGE][IMAGE][IMAGE]66

68

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'll write this out in equation form

[IMAGE][IMAGE]86

88

X + (R-1) + (P(R-1))

[IMAGE]

X + P (R-1)



[IMAGE]




[IMAGE]

X + (R - 1)

X

[IMAGE]17

20

This number is the starting number + the top right corner + the bottom
left corner so I'll write this out in equation form



47 is the starting number + 30 so I need to find a way to add 30

[IMAGE][IMAGE]

X + (R-1) + (P(R-1))

[IMAGE]47

50

[IMAGE]

X + P (R-1)



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.

[IMAGE]


X

X+(R-1)

These are the equations that I have figured out for each corner.

These are the equations that I have figured out for each corner.

[IMAGE]

[IMAGE]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 6x6 square inside a 10x10 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

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

10 x 31 = 310

N is the width

M is the height

X is the starting number

1 x 40 = 40

= 270

I'm now going to see whether the corners have any algebraic relation
to each other.

[IMAGE]

[IMAGE]




1

10

[IMAGE][IMAGE]21

30

X + (N-1) + (M-1)N

21 is the starting number + 20 so I need to find a way to add twenty.

[IMAGE][IMAGE]

X + (M-1)N



[IMAGE]

[IMAGE]




1

10

[IMAGE][IMAGE]41

50

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

[IMAGE]

[IMAGE]

[IMAGE]

41 is the starting number + 40 so I need to find a way to add forty.

X + (M-1)N




X

X+(N-1)

These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]X+(M-1)N

X+(N-1) + (M-1)N

These are the equations that I have figured out for each corner.

(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 7x5 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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

7 X 29 = 203

1 X 35 = 35

Difference = 168

My equation is correct

Rectangles inside rectangles

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

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

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

A is the length of the outer grid

B is the height of the outer grid

C is the length of the inner grid

D is the height of the outer grid

X is the starting number



[IMAGE]

50 is the starting number + 3 so I need to find a way to add three.

X + (C-1)




[IMAGE][IMAGE]47

50

[IMAGE][IMAGE]62

65

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

X + (C-1) + A(D-1)



[IMAGE]

[IMAGE]



[IMAGE]

[IMAGE]




[IMAGE]67

72

X + (C-1) + A(D-1)

[IMAGE][IMAGE]97

102

[IMAGE]

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

97 is the starting number + 30 so I need to find a way to add thirty.



95 is the starting number + 4 so I need to find a way to add four.




[IMAGE]

X + (C-1)

X

[IMAGE][IMAGE]91

95

[IMAGE][IMAGE]106

110

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

106 is the starting number + 15 so I need to find a way to add
fifteen.

[IMAGE]

X + (C-1) + A(D-1)



X

[IMAGE]




[IMAGE]

28 is the starting number + 2 so I need to find a way to add two.

X + (C-1)

[IMAGE]26

28

[IMAGE][IMAGE]41

43

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

[IMAGE]

X + (C-1) + A(D-1)

X + A(D-1)




These are the equations that I have figured out for each corner.

[IMAGE]X

X+(C-1)

These are the equations that I have figured out for each corner.

These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]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 6x5 rectangle a 3x2 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.

Text Box: X is the starting number. N is the length. M is the height. S is the arithmetic progression.
Patterns

What happens when I change the pattern inside a rectangle?

Arithmetic Progressions

8 is the starting number + 6 so I need to find a way to add six.

[IMAGE]



[IMAGE]

[IMAGE]




2

8

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically



[IMAGE][IMAGE][IMAGE]18

24

X + (SN(M-1) + S(N-1))

[IMAGE]

[IMAGE]




6 is the starting number + 4 so I need to find a way to add four.

[IMAGE]

X + S(N-1)

[IMAGE]2

6

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically



[IMAGE]

X + (SN(M-1) + S(N-1))

[IMAGE][IMAGE]14

18

[IMAGE]

X + SN(M-1)



14 is the starting number + 12 so I need to find a way to add twelve.

12 is the starting number + 9 so I need to find a way to add nine.

[IMAGE]




X + S(N-1)

[IMAGE]

X

[IMAGE]3

12

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically



[IMAGE]

X + (SN(M-1) + S(N-1))

[IMAGE][IMAGE]39

48

X + SN(M-1)



[IMAGE]




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.

[IMAGE]


X is the starting number.

N is the length of the rectangle.

M is the height of the rectangle.

S is the arithmetic progression you are using

X

X+S(N-1)

These are the equations that I have figured out for each corner.



These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]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 6x6 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?

X is the starting number.

N is the length of the rectangle.

M is the height of the rectangle.




16 is the starting number + 14 so I need to find a way to add
fourteen.

[IMAGE]

XN

[IMAGE][IMAGE]2

16

This number is the top right corner X the bottom left corner divided
by the starting number.

X



[IMAGE]

(XN) (XN(M-1))

[IMAGE][IMAGE]8192

65536

X(XN(M-1))

[IMAGE]



[IMAGE]

[IMAGE]




2

8

[IMAGE]

This number is the top right corner X the bottom left corner divided
by the starting number.

[IMAGE][IMAGE]16

64

(XN) (XN(M-1))

[IMAGE]

[IMAGE]




XN

X

[IMAGE][IMAGE]3

27

This number is the top right corner X the bottom left corner divided
by the starting number.

[IMAGE][IMAGE][IMAGE]2187

19683

(XN) (XN(M-1))

[IMAGE]




[IMAGE]

These are the equations that I have figured out for each corner.

X

XN

These are the equations that I have figured out for each corner.

These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]X(XN(M-1))

XN (XN(M-1))

X is the starting number.

N is the length of the rectangle

M is height of the rectangle.



(XN) (X(XN(M-1))) = X(XN) + XN (XN(N-1))

(X) (XN (XN(M-1))) = X(XN) + XN (XN(N-1))

Difference = 0

Check

For a 3x3 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

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

62

64

66

68

70

72

74

76

78

80

82

84

86

88

90

92

94

96

98

100

102

104

106

108

110

112

114

116

118

120

122

124

126

128

130

132

134

136

138

140

142

144

146

148

150

152

154

156

158

160

162

164

166

168

170

172

174

176

178

180

182

184

186

188

190

192

194

196

198

200

A is the length of the outer grid.

B is the height of the outer grid.

C is the length of the inner grid.

D is the height of the inner grid.

X is the starting number.

S is the arithmetic progression.

138 is the starting number +2 so I need to find a way to add two.

X

[IMAGE]




[IMAGE][IMAGE]136

138

X + S(C-1) + SA(D-1)

[IMAGE][IMAGE]156

158

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

[IMAGE]

156 is the starting number +20 so I need to find a way to add twenty.



[IMAGE]

X




X + A(C-1)

[IMAGE][IMAGE]24

28

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

[IMAGE][IMAGE]44

48

[IMAGE]

[IMAGE]

X + S(C-1) + SA(D-1)

X + SA(D-1)



[IMAGE]

[IMAGE]




X + S(C-1)

[IMAGE]88

92

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically



[IMAGE]

X + S(C-1) + SA(D-1)

[IMAGE][IMAGE]128

132

[IMAGE]

128 is the starting number +40 so I need to find a way to add fourty.

X + SA(D-1)



[IMAGE]

[IMAGE]




[IMAGE]12

20

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically



[IMAGE][IMAGE][IMAGE]52

60

X + SA(D-1)

[IMAGE]

X + S(C-1) + SA(D-1)




These are the equations that I have figured out for each corner.

[IMAGE]

A is the length of the outer grid.

B is the height of the outer grid.

C is the length of the inner grid.

D is the height of the inner grid.

S is the arithmetic progression.

X is the starting number.

X

X + S(C-1)

These are the equations that I have figured out for each corner.



These are the equations that I have figured out for each corner.

[IMAGE][IMAGE]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 6x4 outer grid with an
arithmetic progression of 5 the difference of the product of the
opposite corners inside an inner grid of 3x2 will equal 5x6(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

21

22

23

24

25

26

27

28

29

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

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.

[IMAGE]

[IMAGE]




212

213

[IMAGE]

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

NX + (C-1) + A(D-1)

[IMAGE][IMAGE]222

223

[IMAGE]



[IMAGE]

[IMAGE]




244

245

246

247

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

[IMAGE][IMAGE]254

255

256

257

254 is the starting number X 210 so I need to find a way to multiply
the start number by 210.

[IMAGE][IMAGE]

NX + A(D-1)

NX + (C-1) + A(D-1)



276 is the starting number X 24 so I need to find a way to multiply
the start number by 24.

[IMAGE]




[IMAGE]

NX + (C-1)

[IMAGE]272

273

274

275

276

This number is the starting number + the top right corner + the bottom
left corner so I'm going to write this algebraically

NX + A(D-1)

282

283

284

285

286

[IMAGE][IMAGE]

NX + (C-1) + A(D-1)

[IMAGE][IMAGE]292

293

294

295

296

292 is the starting number X 220 so I need to find a way to multiply
the start number by 220.



These are the equations that I have figured out for each corner.

[IMAGE]NX

NX+(C-1)

These are the equations that I have figured out for each corner.

These are the equations that I have figured out for each corner.



[IMAGE][IMAGE]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 7x3 outer grid a 4x2 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

X is the starting number.

N is the length of the rectangle.

M is the height of the rectangle.

What would happen if I spiralled into the centre?

[IMAGE]

[IMAGE]




6 is the starting number + 5 so I need to find a way to add five.

1

2

3

4

5

6

20

21

22

23

24

7

11 is the starting number +10 so I need to find a way to add ten.

16 is the starting number +15 so I need to find a way to add fifteen.

19

32

33

34

25

8

[IMAGE][IMAGE]18

31

36

35

26

9

X+(N-1)+(M-1)

X+2(N-1)+(M-1)

[IMAGE][IMAGE]17

30

29

28

27

10

16

15

14

13

12

11

[IMAGE]

[IMAGE]




1

9

[IMAGE]

11 is the starting number +10 so I need to find a way to add ten.

X+(N-1)+(M-1)

[IMAGE][IMAGE]19

11

X+2(N-1)+(M-1)

[IMAGE]



[IMAGE]

X




[IMAGE]

10 is the starting number +2 so I need to find a way to add two.

[IMAGE]8

10

[IMAGE]

14 is the starting number +6 so I need to find a way to add six.

[IMAGE]

12 is the starting number +4 so I need to find a way to add four.

X+(N-1)+(M-1)

X+2(N-1)+(M-1)

[IMAGE][IMAGE]14

12

[IMAGE]

These are the equations that I have figured out for each corner.

X

X+(N-1)

[IMAGE]

These are the equations that I have figured out for each corner.

[IMAGE]

These are the equations that I have figured out for each corner.

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 7x5 rectangle with a starting number of 3 I predict that the
difference will be

2x3(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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"Opposite Corners Investigation." 123HelpMe.com. 19 Apr 2014
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