# Number grid

Extracts from this document...

Introduction

Example 1 12 13 22 23 12 x 23 = 276 13 x 22 = 286 Difference of 10 Example 2 14 15 24 25 14 x 25 = 350 15 x 24 = 360 Difference of 10 Prediction It appears to be a safe prediction that in a two by two square the difference will always be 10. Algebraic Explanation I will assign a letter to the first number in the 2 x 2 square, n. The next number to the right will therefore be n+1, the number directly below it n+10. The number diagonally across from it will be n+11. I will multiply the corner numbers, as shown in the above examples. Top Left hand corner x bottom right hand corner = n(n+11) = n? + 11n Top right hand corner x bottom left hand corner = n? +1n+10n+10 n? +11n+10 (n? +11n+10) - (n? + 11n) = 10 Therefore the difference between the multiplied corner numbers will always be 10. I believe that it would be interesting to look at a 3x3 number square on a 100 grid. I will take a 3x3 square on a 100 square grid and multiply the two corners. I will then look at the relationship between the two results, by discovering the difference. Example 17 18 19 27 28 29 37 38 39 17 x 39 = 663 19 x 37 = 703 Difference of 40 I am fairly sure that in a ...read more.

Middle

predict that a 5 x 5 square will have a difference of 160 because the differences of the differences increase by 20 each time e.g. from 30 to 50. Therefore it would make sense for 50 to go up to 70, add this onto the 90 which makes 160, which is my prediction for the 5 x 5 square. Proving the prediction: 56 57 58 59 60 66 67 68 69 70 76 77 78 79 80 86 87 88 89 90 96 97 98 99 100 56 x 100 = 5600 60 x 96 = 6760 Difference of 160. Working out a general equation for the difference between the two corners multiplied together with any number of sides on the square Considering all that the 10 times table contained all of the numbers (10, 40, 90, 160) then I will assume the equation has a x10 in it, and I will therefore remove the 0 from each of the numbers. This leaves 1, 4, 9, and 16. These are all square numbers. It is noticeable that the numbers are the square of the side of square minus 1. This means that b-12 makes sense. I will encompass this with the x10 so it becomes: 10(b-1)2 Testing the equation I am going to use the example of a 4 x 4 square; I am already aware that the difference is 90. ...read more.

Conclusion

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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 Example1 70 71 81 82 70 x 82 = 5740 71 x 81 = 5751 Difference of 11 Example 2 43 44 54 55 43 x 55 = 2365 44 x 54 = 2376 Difference of 11 It is clear that in a 4 x 4 box, in an 11 x 11 grid, the difference will always be 11. Conclusion I conclude that in this investigation, I have created numerous formulas which demonstrated the differences between corner numbers, and how they relate to the side numbers in the box. I have also changed the size of the original grid, but I was too short of time to create a formula for a 4 x 4 box in a 11 x 11 grid, which I would have done, had I worked on this investigation over a lengthier period. Top of Form ?? ?? ?? ?? 1 ...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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