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

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.

Get quality help now

writer-Charlotte

Verified writer

4.7 (348)

“ Amazing as always, gave her a week to finish a big assignment and came through way ahead of time. ”

+84 relevant experts are online

Hire writer

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

?

?

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.

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?

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

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 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

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 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

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 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.

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 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?

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 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

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 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.

212

213

222

223

244

245

246

247

254

255

256

257

272

273

274

275

276

282

283

284

285

286

292

293

294

295

296

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 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

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 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.