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  • Level: GCSE
  • Subject: Maths
  • Word count: 3166

Number Grid

Extracts from this document...

Introduction

NUMBER GRID Look at this number 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 10 x 10 grid * A box is drawn round four numbers. * Find the product of the top left number and the bottom right number in this box. * Do the same with the top right and bottom left numbers. * Calculate the difference between these products. Investigate further. The purpose of this investigation is to prove or disprove that there is a correlation between the products of corner numbers of any size box within any size grid. I shall calculate the diagonal difference (d) for a box. There are two ways to calculate the difference between the products: V W Y Z i. W x Y - V x Z ii. V x Z - W x Y I shall begin with the 10 x 10 grid, as shown above, and a 2 x 2 box. 2 x 2 box #1 I have substituted numbers for the letters in my two formulae. 1 2 11 12 i. d = 2 x 11 - 1 x 12 = 22 - 12 = 10 ii. d =1 x 12 - 2 x 11 = 12 - 22 = -10 d = +/-10 I will use formula i, because formula ii creates negative numbers, which could make my calculations more complex than necessary. ...read more.

Middle

x 8 2 x 7 2 x 6 2 x 5 2 x 4 2 x 3 3 x 10 3 x 9 3 x 8 3 x 7 3 x 6 3 x 5 3 x 4 4 x 10 4 x 9 4 x 8 4 x 7 4 x 6 4 x 5 5 x 10 5 x 9 5 x 8 5 x 7 5 x 6 6 x 10 6 x 9 6 x 8 6 x 7 7 x 10 7 x 9 7 x 8 8 x 10 8 x 9 9 x 10 This list shows 36 possible sizes. If the orientation of the rectangle produces a different value for d, then there will be 72 possible sizes. My first task is to ascertain if the orientation of the rectangle affects the diagonal difference. 3 x 2 rectangle #1 1 2 3 11 12 13 d = 3 x 11 - 1 x 13 d = 33 - 13 d = 20 3 x 2 rectangle #2 12 13 14 22 23 24 d = 14 x 22 - 12 x 24 d = 308 - 288 d = 20 3 x 2 rectangle #3 88 89 90 98 99 100 d = 90 x 98 - 88 x 100 d = 8820 - 8800 d = 20 2 x 3 rectangle #1 1 2 11 12 21 22 d = 2 x 21 - 1 x 22 d = 42 - 22 d = 20 2 x 3 rectangle #2 12 13 22 23 32 33 d = 2 13 x 32 - 12 x 33 d = 416 - 396 d = 20 2 x 3 rectangle #3 79 80 89 90 99 100 d = 80 x 99 - 79 x 100 d = 7920 - 7900 d = 20 The results indicate that the diagonal difference is not dependant on the orientation of the rectangle. ...read more.

Conclusion

2 or d = G(x - 1) (x - 1) As there are two different values for the sides of a rectangle, I think that the formula would be: d = G(x - 1) (y - 1) To test this formula, I have calculated the difference for three different sized boxes on three different sized grids. 3 x 8 rectangle on a 9 x 9 grid 1 3 64 66 d = 3 x 64 - 1 x 66 d = 192 - 66 d = 126 +2 +7G a a + 2 a + 7G a +7G + 2 d = (a + 2) (a + 7G) - a (a + 7G + 2) = a2 + 7aG + 2a + 14G - a2 - 7aG - 2a = 14G, where G = 9, d =126 4 x 6 rectangle on a 7 x 7 grid 1 4 36 39 d = 4 x 36 - 1 x 39 d = 144 - 39 d = 105 +3 +5G a a + 3 a + 3G a + 5G + 3 d = (a + 3) (a + 5G) - a (a + 5G + 3) = a2 + 5aG + 3a + 15G - a2 - 5aG - 3a = 15G, where G = 7, d = 105 2 x 4 rectangle 0n a 5 x 5 grid 1 2 16 17 d = 2 x 16 - 1 x 17 d = 32 - 17 d = 15 +1 +3G a a + 1 a + 3G a + 3G + 1 d = (a + 1) (a + 3G) - a (a + 3G + 1) = a2 + 3aG + a + 3G - a2 - 3aG -a = 3G, where G = 5, d = 15 To conclude, I have found a master formula to find the difference of any sized box which is defined thus: x x y , where x can be equal to y on grid with any number of columns (G). d = G(x - 1) (y - 1) - 1 - ...read more.

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