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Introduction

Investigating when pairs of diagonal corners are multiplied and subtracted from each other. Introduction: In this coursework I shall be investigating when pairs of diagonal corners are multiplied and subtracted from each other. To do this I shall take square grids and then select 3 square boxes from each grid and multiply the opposite corners then find the difference, then I will change the box size, to try and find a pattern or formula. Then I will change the grid size and then once again take boxes of different sizes and multiply the opposite corners and find the difference. I will try and find a formula for and square box on and size square grid. Then I shall investigate further by using rectangle boxes instead of square. 34 x 45 = 1530 35 x 44 = 1540 Difference = 10 68 x 79 = 5372 69 x 78 = 5382 Difference = 10 81 x 92 = 7452 82 x 91 = 7462 Difference = 10 In a 2 x 2 box on a 10 x 10 grid the difference is 10. Algebraic Method x x + 1 x + 10 x + 11 x(x + 11) = x� + 11x (x + 1)(x + 10) = x� + 10 + x +10x = x� + 11x + 10 x� + 11x - x� + 11x + 10 = 10 1 x 23 = 23 3 x 21 = 61 Difference = 40 36 x 58 = 2088 38 x 56 = 2128 Difference = 40 73 x 95 = 6935 75 x 93 = 6975 Difference = 40 In a 3 x 3 box on a 10 x 10 grid the difference is 40. Algebraic Method x x + 2 x + 20 x + 22 x(x + 22) = x� + 22x (x + 2)(x + 20) = x� + 20x + 2x + 40 = x� + 22x + 40 x� + 22x - x� + 22x + 40 = ...read more.

Middle

N = Any Number (b -1) = Box size - 1 Let x = (b - 1) n n + x n + 8x n + 9x n(n + 9x) = n� + 9xn (n + 1)(n + 8) = n� + 8xn + xn + 8x� = n� + 9xn + 8x� n� + 9xn - n� + 9xn + 8x� = 8x� 8x� = 8(b-1) � I will now do the same process on an 11 x 11 grid to see if there is a pattern. 14 x 26 = 364 15 x 25 = 375 Difference = 11 28 x 40 = 1120 29 x 39 = 1131 Difference = 11 76 x 88 = 6688 77 x 87 = 6699 Difference = 11 In a 2 x 2 box on an 11 x 11 grid the difference is 11. Algebraic Method x x + 1 x + 11 x + 12 x(x + 12) = x� + 12x (x + 1)(x + 11) = x� + 11x + x + 11 = x� + 12x + 11 x� + 12x - x� + 12x + 11 = 11 5 x 29 = 145 7 x 27 = 189 Difference = 44 42 x 66 = 2772 44 x 64 = 2816 Difference = 44 89 x 113 = 10057 91 x 111 = 10101 Difference = 44 In a 3 x 3 box on an 11 x 11 grid the difference is 44. Algebraic Method x x + 2 x + 22 x + 24 x(x + 24) = x� + 24x (x + 2)(x + 22) = x� + 22x + 2x + 44 = x� + 24x + 44 x� + 24x - x� + 24x + 44 = 44 4 x 40 = 160 7 x 37 = 259 Difference = 99 78 x 114 = 8892 81x 111 = 8991 Difference = 99 74 x 110 = 8140 77 x 107 = 8239 Difference = 99 In a 4 x 4 box on an 11 x 11 grid the difference is 99. ...read more.

Conclusion

= n� + 9xn + ny (n + y)(n + 9x) = n� + 9xn + ny + 9xy n� + 9xn + ny - n� + 9xn + ny + 9xy = 9xy 9xy = 9(L - 1) (W - 1) I have noticed a pattern in my formulas. The number that comes before the brackets is the same as the grid size. So I can change that number to G when G = Grid Size. So the formula for any box size on any grid size: G (L - 1)(W - 1) Proving Formula: n n+(W - 1) n + G(L - 1) n + G(L -1) + (W - 1) N = Any Number (L - 1) = Length - 1 (W - 1) = Width - 1 Let x = (L - 1) Let y = (W - 1) n n + y n + Gx n + Gx + y n(n + Gx + y) = n� + nGx + ny (n + y)(n + Gx) = n� + nGx + ny + Gxy n� + nGx + ny - n� + nGx + ny + Gxy = Gxy Gxy = G (W- 1)(L - 1) Conclusion Formula Description 10(b - 1) � This calculates the difference of any square box on a 10 x 10 grid. 9(b - 1) � This calculates the difference of any square box on a 9 x 9 grid. 8(b - 1) � This calculates the difference of any square box on an 8 x 8 grid. 11(b - 1) � This calculates the difference of any square box on an 11 x 11 grid. G(b - 1) � This calculates the difference of any square box on any grid. 10(L - 1)(W - 1) This calculates the difference of any rectangle box on a 10 x 10 grid. 9(L - 1)(W - 1) This calculates the difference of any rectangle box on a 9 x 9 grid. G(L - 1)(W - 1) This calculates the difference of any rectangle box on any grid. Limitations: ...read more.

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