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Number Grids

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Introduction

Number Gridss 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 Use the following rule: Find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and top right numbers in the square. Calculate the difference between these numbers. INVESTIGATE! To start this assignment I randomly selected an area in the matrix to put boxes of size; (2x2) ...read more.

Middle

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 Z 92 93 94 95 96 97 98 99 Q I came up with: X=X Y=X + (n-1) Z=X + ((n-1) x 10) Q=X + ((n-1) x 10) + (n-1) I used this to formulate the equation: X x X+11(n-1)) - ((X+ (n-1)X+10(n-1))) This cancels down to: (X2+11X(n-1))-(X2+11X(n-1) +10(n-1)2) To work out my overall formula I will subtract the first part of the Formula from the other: X2+11X(n-1) - X2+11X(n-1)) +10(n-1)2 0 + 0 + 0 +10(n-1)2 =10(n-1)2 Rectangles Solving the puzzle of rectangles in a grid came quite quickly to me as I realized that n was just for use in a square because the dimensions were the same. I labeled the vertical length now as m, but keeping the horizontal length as n. I adapted the equation from 10(n-1)2 into 10(n-1)(m-1). ...read more.

Conclusion

However, it should work with a rectangular grid as well. I tried a 3x4 inner grid in a 6x7 outer 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 (3-1)(4-1)7 =2x3x7 =42 The formula still works. 4x23 = 92, 2x25 = 50, 92-50 = 42 so I have found a formula which will work whatever of the size, inner or outer grid. Conclusion I have now found three equations, from these I can find any size of rectangle or square in any size of number grid. They are: * 10(n-1)2, for any square inside a 10x10 grid * 10(n-1)(m-1), for any rectangle inside a 10x10 grid * g(n-1)(m-1), for any rectangle or square inside any grid Josef Jeffrey Math's G.C.S.E coursework ...read more.

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