# Number grid

Extracts from this document...

Introduction

Samantha Whittaker Number 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 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 Pick out 2x2 squares Multiply the diagonals Find the difference 34 35 44 45 INVESTIGATE 27 28 37 38 82 83 92 93 68 69 78 79 9 10 19 20 First number Second number Third number Fourth number 1stx4th 2ndx3rd Difference 34 35 44 45 1530 1540 10 27 28 37 38 1026 1036 10 82 83 92 93 7626 7636 10 68 69 78 79 5372 5382 10 9 10 19 20 180 190 10 I have put my results into a table so that they are easier to analyse and compare. What I have found is; when you take a 2x2 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 10. 36 37 46 47 If this rule is correct, then by using this grid: I predict that the difference between the Product of 36 and 47 compared with that of 37 and 46 will equal 10. 36x47=1692 37x46=1702 Difference=10 My prediction was correct as the difference between 36x47 (1692), and 37x46 (1702) was 10. To prove that this theory will work for any 2x2 grid from a 10x10 number square, I am going to express it as an algebraic equation. X X+1 X+10 X+11 (X)(X+11)=(X+1)(X+10) X�+11X=X�+11X+10 The equation is set out with the first half being the top left multiplied by the bottom right and the second half being the top right multiplied by the bottom left. ...read more.

Middle

X X+1 X+10 X+11 X+20 X+21 (X)(X+21)=(X+1)(X+20) X�+21X=X�+21X+20 The equation is set out with the first half being the top left multiplied by the bottom right and the second half being the top right multiplied by the bottom left. The 20 that is underlined in the above equation, proves that the latter part of the equation will always be 20 more than the first half of the equation. I shall now investigate grids of 5x3 from a 10x10 number square. 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 42 43 44 45 46 52 53 54 55 56 62 63 64 65 66 76 77 78 79 80 86 87 88 89 90 96 97 98 99 100 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 35 36 37 38 39 45 46 47 48 49 55 56 57 58 59 First number Fifth number Eleventh number Fifteenth number 1stx15th 5thx11th Difference 1 5 21 25 25 105 80 42 46 62 66 2772 2852 80 76 80 96 100 7600 7680 80 6 10 26 30 180 260 80 35 39 55 59 2065 2145 80 I have put my results into a table so that they are easier to analyse and compare. What I have found is; when you take a 5x3 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 80. 54 55 56 57 58 64 65 66 67 68 74 75 76 77 78 If this rule is correct, then by using this grid: I predict that the difference between the product of 54 and 78 compared with that of 58 and 74 will equal 80. 54x78=4212 58x74=4292 Difference= 4292-4212=80 My prediction was correct as the difference between 54x78 (4212), and 58x74 (4292) was 80. To prove that this theory will work for any 5x3 grid from a 10x10 number square, I am going to express it as an algebraic equation. ...read more.

Conclusion

According to my calculations, the difference between 49x177 and 57x169 should be 960. 49x177=8673 57x169=9633 Difference=9633-8673=960 My prediction was correct; therefore the formula I constructed will apply in all cases of squares taken from number squares. I will now try and devise a formula to find out the difference when multiplying the diagonals of rectangles taken from a number square. Side of original grid Length of extracted grid Width of extracted grid Difference 10 2 3 20 10 5 3 80 10 2 5 80 5 4 3 30 5 5 2 20 5 3 4 30 7 2 5 28 7 6 3 70 7 5 2 28 Let A be the side of the original grid, B be the length of the extracted grid, C be the width of the extracted grid and D be the difference. D= (B-1)(C-1) x A This means that the difference between the products of each diagonal on a square taken from a number grid will be the length of the extracted grid minus one times the width of the extracted grid minus one then multiplied by the side of the original number square. CHECK 92 93 94 95 96 97 98 99 100 101 102 103 104 105 107 108 109 110 111 112 113 114 115 116 117 118 119 120 122 123 124 125 126 127 128 129 130 131 132 133 134 135 137 138 139 140 141 142 143 144 145 146 147 148 149 150 152 153 154 155 156 157 158 159 160 161 162 163 164 165 167 168 169 170 171 172 173 174 175 176 177 178 179 180 The above is a 14x6 grid taken from a 15x15 number square. According to my calculations, the difference between 92x180 and 105x167 should be 975. 92x180=16560 105x167=17535 Difference=17535-16560=975 My prediction was correct; therefore the formula I constructed will apply in all cases of rectangles taken from number squares. ...read more.

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