# Opposite Corners Investigation

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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

## 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)

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

(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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