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Introduction

Number Grids 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 This grid shows the numbers one to one hundred. The two of the corners diagonally opposite each other on a grid like this can be multiplied together to make a number. The other to corners can be multiplied to give a second number; there is a difference between the two figures. This difference is the interesting figure. We can compare the difference between grids of different sizes. 1 2 11 12 Let us start of small with a 2x2 grid. We can work out the difference using the long method first. 1x12 = 12 2x11 = 22 22-12 = 10. Therefore, ten is the difference of our first example. Quite easy you would think, but nerveless there are three sums, And, as the numbers get bigger the sums get harder. N N+1 N+10 N+11 So how can we get around this? ...read more.

Middle

32 33 34 35 36 37 41 42 43 44 45 46 47 51 52 53 54 55 56 57 61 62 63 64 65 66 67 10(7-1)� = 10(6)� =360 So we have our difference and with one easy calculation. But lets check it using the normal method. N N+1 N+2 N+3 N+4 N+5 N+6 N+10 N+11 N+12 N+13 N+14 N+15 N+16 N+20 N+21 N+22 N+23 N+24 N+25 N+26 N+30 N+31 N+32 N+33 N+34 N+35 N+36 N+40 N+41 N+42 N+43 N+44 N+45 N+46 N+50 N+51 N+52 N+53 N+54 N+55 N+56 N+60 N+61 N+62 N+63 N+64 N+65 N+66 (n+6)(n+60) =n�+60n+6n+360 So our formula works. And the difference for a 7x7 grid is 360. Now we move on to Rectangular grids. 4 5 14 15 24 25 34 35 44 45 54 55 1 2 3 11 12 13 These are both rectangles but we are going to start buy looking at a regular pattern of rectangle size. 2x3, 3x4, 4x5. And so on. Let us start with a 2x3 grid. First, let us use the basic method to find the difference. 1x13 = 13 3x11 = 33 N N+1 N+2 N+10 N+11 N+12 33-11 = 20. So our difference is 20. But will our formulas from the square grids work? 10(n-1) will not work as it dose not take into account the two different lengths of the sides. However, our other formula may work let us try it. ...read more.

Conclusion

Firstly, we will try a 3x3 (red) inner grid in a 7x7 (black) 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 43 44 45 46 47 48 49 (3-1)(3-1)7 =2x2x7 =28 so lets check it. (n+2)(n+14) n�+14n+2n+28 so our formula works. However, will it work with a rectangular grid. Lets try 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. Nevertheless, let us check just in case. 4x23 = 92, 2x25 = 50, 92-50 = 42 so we have found the formula which will work foe any size inner or outer grid. Conclusion I have found three main formulas: For squares is a 10x10 grid the formula 10(n-1)�, can be used. For rectangles (also works for squares) in a 10x10 grid the formula (w-1) (l-1) 10 can be used For rectangles (and squares) in any sized grid the formula (w-1) (l-1) g can be used. With the third of these three formulas it is possible to work out the difference on any sized square or rectangle in any sized grid. In a way, it is the concluding formula. Without moving into different or thee dimensional shapes. 1 ????????????? ...read more.

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