# Maths GCSE Investigation - T Numbers

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Introduction

Maths GCSE Investigation - T Numbers Introduction In a number grid, a T-shape can be drawn outlining five numbers. The sum of all the numbers within the T-shape is the T-total, and the number at the base of the T is the T-number. 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 In this 5 by 5 grid example, the T-number is 12, and the T-total is 1+2+3+7+12, which is 25. My task is to investigate the relationship between the T-number and the T-total for T-shapes within the number grid. The factors I can change which might change the pattern of numbers are: * Grid size * Direction of T-shape (with the T pointing N/S/E/W) * Translating the T-shape within a number grid to investigate patterns. Investigation#1-upright T-shape First of all, I am going to focus on the T-shape pointing S, the upright T: 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 I first thought that any T-shape could be created if the T-number was known, as the rest of the numbers in the T-shape are related to the T-number. For example, in the above 6 by 6 square, a T-shape is selected: T-number=21 T-total=21(T-number) ...read more.

Middle

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 The upright T-shape and the upside-down T-shape had an opposite in their formulas: -7Z and +7Z. Due to this fact, I predict that the formula for a sideways T-shape pointing W would be 5X+7, opposite to the sideways T-shape pointing E's formula of 5X-7. To test my prediction, I will apply it to the above 9 by 9 grid: T-total = 5X+7 = 5*51+7 = 262 51+52+53+44+62 = 262 My formula works, so 5T+7 is the formula for a sideways T-shape pointing W. Summary From the above four investigations, I can summarise that: * The formula for an upright T-shape in ANY grid size is: T-total = 5X-7Z * The formula for an upside-down T-shape in ANY grid size is: T-total = 5X+7Z * The formula for a sideways T-shape pointing E in ANY grid size is: T-total = 5X-7 * The formula for a sideways T-shape pointing W in ANY grid size is: T-total = 5X+7 Further investigations There are several different variables which I can change to extend this investigation: * Investigate in 3 dimensions * Translation of T-shape * Rotation of T-shape * Reflection of T-shape * Enlargement of T-shape 3D investigations A basic 3D cube can be made from small cubes, each representing one number. ...read more.

Conclusion

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 Applying the rules previously, the T-number in the above shape should be 35. The T-total is 105. The formula is: T-total = X(35) + X-Z+1(25) + X-2Z+2(15) + X-3Z+1(3) + X-Z+3(27) = 5X-7Z+7 = 5*35 - 7*11 + 7 = 105 3+15+25+27+35 = 105 From the derivation of my formula, I have proved that the T-shape and its number are in a fixed position, so it will work wherever it is placed. Test formula for 6 by 6 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 T-total = 5X-7Z+7 = 5*20-7*6+7 = 65 3+10+15+17+20 = 65 My formula works for the 6 by 6 grid. So the formula for a T-shape rotated through 45� is: T-total = 5X-7Z+7 N.B. Since this investigation is based on patterns derived from a diagram, ALL the formulas derived only apply in cases which the T-shape itself fits into the grid. ?? ?? ?? ?? Qiming Liu 10SM 09/05/2007 ...read more.

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