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Number grids.

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Introduction

Number grids. By Nicole Highfield. The aim of this investigation is to find the differences of N x N squares that are inside a 10 x 10 grid (shown bellow) then to see if there is a rule or pattern that will connect the size of the N x N to the size. To be able to find the difference of an N x N square, I will start at the smallest one, which is 2 x 2. The corners have to be multiplied together in a diagonal way. My 2 x 2 square is: - 12 x 23 = 276 13 x 22 = 286 Then to find the difference I have to minus the smallest answer from the biggest one. This is the to find the differences in all squares. 286 - 276 = 10 Using 2 x 2 array (square) of number I have found a difference between the two corners, this is a difference of ten. I will check this by doing some more 2 x 2 arrays, and work my way to bigger squares so that the investigation does not get complicated. 88 89 98 99 88 x 99 = 8712 89 x 98 = 8722 8722 - 8712 = 10 55 56 65 66 55 x 66 = 3630 56 x 65 = ...read more.

Middle

74 75 76 84 85 86 94 95 96 74 x 96 = 7104 94 x 76 = 7144 7144 - 7104 = 40 18 19 20 28 29 30 38 39 40 18 x 40 = 720 20 x 38 = 760 760 - 720 = 40 1 2 3 11 12 13 21 22 23 1 x 23 = 23 3 x 21 = 63 63 - 23 = 40 I am now going to check that the difference is 40 by using the x-box and algebraic formula. The difference is that the x-box will change due to the increase in the array (this will happen every time I increase the arrays). This is the changed x-box for the new the array. X X+1 X+2 X+10 X+11 X+12 X+20 X+21 X+22 X(X+22) = X +22X (X+2) (X+20) = X +40+2X +20 X +40+22X X +22X___ 40_______ The difference for the 3 x 3 array is 40 so there is defiantly an increase of 30 from the 2 x 2 to the 3 x 3. Now my array will increase even more to a 4 x 4 array, and will get bigger until I find some kind of pattern. ...read more.

Conclusion

23 24 25 26 27 33 34 35 36 37 43 44 45 46 47 53 54 55 56 57 63 64 65 66 67 23 x 67 =1541 27 x 63 = 1701 1701 - 1541 = 160 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 36 37 38 39 40 46 47 48 49 50 6 x 50 = 300 10 x 46 = 460 460 - 300 = 160 Now I need to check with the x-box that the answer is 160. X X+1 X+2 X+3 X+4 X+10 X+11 X+12 X+13 X+14 X+20 X+21 X+22 X+23 X+24 X+30 X+31 X+32 X+33 X+34 X+40 X+41 X+42 X+43 X+44 X(X+44X) = X +44X (X+4) (X+40) = X +160+4X+40X X +160+44X X +44X____ 160_______ this show me that the x-box and my answer are right the difference is 160, that's an increase of 70. To get a pattern I must do some more so I will go to a 7 x 7 array, this should be big enough to get a good pattern. Then I will be able to predict what a 60 x 60 array is and so on. The next array is 6 x 6. ...read more.

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