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Introduction

Task B: Diagonal Difference Name: Terry Curtis Centre Name: Gaynes School Form: 11s2 Centre Number: 12847 Candidate Number: 7045 Teacher Name: Mr Nash Index Page Page 1: Title Page Page 2: Index Page Page 3: Statement Page 4: Initial Investigation, 3 x 3 grids inside a 8 x 8 grid Page 5: Initial Investigation, 4 x 4 grids inside a 8 x 8 grid Page 6: Initial Investigation, 5 x 5 grids inside a 8 x 8 grid Page 7: Initial Investigation, 2 x 2 grids inside a 8 x 8 grid Page 8: Justifying My Results, form for a 3 x 3 and 4 x 4 grid inside a 8 x 8 gird Page 9: Justifying My Results, form for a 2 x 2 and 5 x 5 gird inside a 8 x 8 grid Page 10: Justifying My Results, prediction for a 6 x 6 grid inside a 8 x 8 grid Page 11: Justifying My Results, justifying my prediction Page 12: Justifying My Results, nth term found for the results of a 8 x 8 grid Page 13: Justifying My Results, formula found for any grid inside a 8 x 8 gird Page 14: Justifying My Results, Formula tested on previous problems Page 15: Further Investigation, 6 x 6 grid Page 16: Further Investigation, 6 x 6 grid Page 17: Justifying My Results, formula found for any gird inside 6 x 6 grid Page 18: Justifying My Results, formula tested for any grid inside a 6 x 6 grid Page 19: Further Investigation, 7 x 7 grid Page 20: Further Investigation, 7 x 7 grid Page 21: Justifying My Results, formula found for any gird inside 7 x 7 grid Page 22: Justifying My Results, formula tested for any grid inside a 7 x 7 grid Page 23: Further Investigation, formula found for any grid inside any grid Page 24: Justifying My Results, formula tested on previous grids of all sizes Page 25 : Justifying My Results, formula tested on ...read more.

Middle

- 8 1 =b2 + 9ab - 9b 2 = a2 + 9ab - 9b + 8a2 - 16a + 8 2 - 1 = b2 + 9ab - 9b + 8a2 - 16a + 8 b2 + 9ab - 9b Justifying My Results I will see if my formula works by testing it on the calculations I have already made. Large grid Small grid Results 8 x 8 2 x 2 8 8 x 8 3 x 3 32 8 x 8 4 x 4 72 8 x 8 5 x 5 128 8 x 8 6 x 6 200 8 x 8 7 x 7 288 a = side of small gird Any 2 x 2 grid inside a 8 x 8 grid: = 8a2 - 16a + 8 = 8 x 22 - 16 x 2 + 8 = 8 Any 4 x 4 grid inside a 8 x 8 gird: = 8a2 - 16a + 8 = 8 x 42 - 16 x 4 + 8 Further Investigation 6 x 6 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 3 x 3 grid : 1 2 3 7 8 9 13 14 15 4 5 6 10 11 12 16 17 18 19 20 21 25 26 27 31 32 33 22 23 24 28 29 30 34 35 36 1 x 15 = 15 4 x 18 = 72 19 x 33 = 627 22 x 36 = 792 3 x 13 = 39 6 x 16 = 96 21 x 31 = 651 24 x 34 = 816 39 - 3 = 24 96 - 72 = 24 651 - 627 = 24 816 - 792 = 24 The diagonal difference for any 3 x 3grid inside a 6 x 6 grid = 24 4 x 4 grid ...read more.

Conclusion

1 x 46 = 46 6 x 41 = 246 246 - 46 = 200 My prediction was right for the diagonal difference for any 6 x 5 grid inside a 10 x 10 square. =cab - cb - ca + c =(10 x 2 x 3) - (10 x 3) - (10 x 2) + 10 =20 Using my formula, I predict that the diagonal difference for any 3 x 2 rectangular grid inside a 10 x 10 grid will be 20. My prediction was right for the diagonal difference for any 3 x 2 rectangular grid inside a 10 x 10 grid. I have found a number of formula's to find the diagonal difference of any grid inside any grid even for the original problem of a 3 x 3 gird inside a 8 x 8 grid and a formula for the any rectangular grid inside any square grid. All of these formulas were found using algebra and all of these formulas have been justified by checking them with old results and by predicting new results and making sure they worked. Below are all the formulas I have found: What for Formula Any grid inside a 6 x 6 grid 6a2 - 12a + 6 Any grid inside a 7 x 7 grid 7a2 - 14a + 7 Any grid inside a 8 x 8 grid 8a2 - 16a + 8 General formula for any grid inside any grid ca2 - 2ca + c Any rectangular grid inside any square grid cab - cb - ca + c Were c = size of one of the sides of the larger grid b = size of one of the sides of the smaller grid a = size of the other side of the smaller grid i.e. c = 8 b = 3 a = 4 This would be the numbers used for a small 3 x 4 grid inside the larger 8 x 8 grid. 1 ...read more.

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