# Investigation into Number Grids.

Extracts from this document...

Introduction

Jonathan Shields 4CH Number Grids Square Grids 1 2 3 4 5 n n+1 n+2 n+3 n+4 11 12 13 14 15 n+10 n+11 n+12 n+13 n+14 21 22 23 24 25 n+20 n+21 n+22 n+23 n+24 31 32 33 34 35 n+30 n+31 n+32 n+33 n+34 41 42 43 44 45 n+40 n+41 n+42 n+43 n+44 The left hand grid (with a black boarder) shows the number grid as it is. The left hand grid (boxed in grey) shows the numbers relative to n. Grid Size Working Answer 2 x 2 (1 x 12) - (2 x 11) -10 3 x 3 (1 x 23) - (3 x 21) -40 4 x 4 (1 x 34) - (4 x 31) -90 5 x 5 (1 x 45) - (5 x 41) -160 So for 2 x 2, where n is the width, and therefore the length of the square. [n�+11n] - [(n+1)(n+10] [n�+11n] - [n�+10n+n+10] 11n + 10n + n +10 = -10 And for 3 x 3. [n�+22n] - [(n+2)(n+20)] [n�+22n] - [n�+20n+2n +40] 22n - 20n - 2n - 40 = -40 Examples using different grids: For 4 x 4. 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 So (11 x 44) - (14 x 41) = - 90 So for any 4 x 4 box the answer will always be - 90. For 5 x 5. ...read more.

Middle

So:-10v�(x-1)(x-1) -10x4�(3-1)(3-1) -10x16(2 x 2) -160 x 4 = -640 If you look at the results table for the grid with a step value 4 we find that this answer is right, as the answer is -640. To make sure: -10v�(x-1)(x-1) -10x4�(8-1)(8-1) -10x16(7x7) -160x49 =-7840 This proves the equation must work for all equations with a different step value. Even where the step value is 1, you could use it but it world just be like saying -10 when u say -10 multiplied by 1. Since -10, I took from the grid which was 10 boxes wide, I shall see if it makes any difference if you change the width of the 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 The results table below shows the results when the different box sizes are used to find an answer from a grid only 5 wide. This is to try and see if that grid width could be used as another variable in the equations. Box size Workings Box Value 2 by 2 (1*7)-(2*6) -5 3 by 3 (1*13)-(3*11) -20 4 by 4 (1*19)-(4*16) -45 5 by 5 (1*25)-(5*21) -80 The results shown in the table of results above are exactly half of what they were for a grid with width of 10. ...read more.

Conclusion

Therefore the formula can be shown as: -wv�(x-1)(y-1) Again where w is the width of the grid, v� is the step value squared, x is the height and y is the height. We can use the following grid to try this formula out. The grid below has a step value of 2 so v� will be 4. 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 I shall use the rectangle highlighted in blue. Using the normal way first: (1 x 101)-(5 x 97)=-384 Now by using the formula: -wv�(x-1)(y-1) -8x2�(3-1)(7-1) -8x4(2x6) -32x12 =-384 Again using the same formula and the same grid, but using the box highlighted in red. Using the normal way: (11 x 125)-(13 x 123) 1375 - 1599 =-224 Using the formula:-wv�(x-1)(y-1) -8x2�(2-1)(8-1) -8x4(1x7) -32 x 7 = -224 Another way of putting the equation -wv�(x-1)(y-1) w=width of grid v�=step value (squared) x=width of box y=height of box You could have the equation: -xy�(c-1)(r-1) x=grid width y�=step value c=columns in box r=rows in box. Any shaped box -xv�(r-1)(c-1) v is the step value n top right hand number r # of rows c # of columns ...read more.

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