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Investigation into T-Totals.

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Introduction

Investigation into T-Totals The 9*9 grid below has a T - shape drawn on it. The sum of the numbers inside the T - shape is called the 'T - Total'. The number in the bottom square of the T - shape is called the 'T - number' (Number in green). 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 So, the T - total for this T - shape is: 1 + 2 + 3 + 11 + 20 = 37 So 37 = T-total The T-number for this shape is 20 (shown in green). Aims: 1) Investigate the relationship between the T-total and the T-number. 2) Use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size. 3) Use grids of different sizes again. Try other transformations and combinations of transformations. ...read more.

Middle

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 As there are 10 in each row, it's obvious that the row above will be 10 less than the row below. So, 68 is 10 less than the T-number 78. If you calculate the whole T - shape, you realise that row 2 is 10 less than row 1, and row 3 is 20 less than row 1. Idea 1 I have decided to see if grid size makes any difference. I have compiled three grids into a table, and am investigating the same theory though numbers in intervals of ten (20, 30-60, 25, 35-65). Grid Size Formula T-Number T-Total 9*9 5N - 7W = T-Total 20 (5*20) - (7*9) = 37 9*9 5N - 7W = T-Total 30 (5*30) - (7*9) = 87 9*9 5N - 7W = T-Total 40 (5*40) - (7*9) = 137 9*9 5N - 7W = T-Total 50 (5*50) - (7*9) = 187 9*9 5N - 7W = T-Total 60 (5*60) - (7*9) = 237 9*9 5N - 7W = T-Total 25 (5*25) ...read more.

Conclusion

2 312 320 12*12 Brown 3 632 Idea 1 I noticed in the graph above that they are all factors of 5. So, I am going to try close up T-Shapes to see if there is a pattern. Prediction: There is a pattern of 5 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 Grid Size T-Number T-Totals Difference between the three T-Shapes 12*12 2 94 5 12*12 3 99 12*12 4 104 12*12 78 383 5 12*12 79 388 12*12 80 393 12*12 130 657 5 12*12 129 652 12*12 128 647 My prediction was correct, as there is a difference of 5 for each new T-Shape. ...read more.

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