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Introduction

David Jacques

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

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)

Middle

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

Conclusion

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.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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