# For my investigation, I will be investigating if there is a relationship between t-total and t-number. I will first try to find a relationship between T-number and T-Total on a 9x9 grid

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Introduction

Maths Coursework

By James Lathey

For my investigation, I will be investigating if there is a relationship between t-total and t-number. I will first try to find a relationship between T-number and T-Total on a 9x9 grid then change the variables such as grid size. I will also be looking at what effect rotation has.

N-7 |

N-9 |

N-8 |

2 |

1 |

3 |

N-9 |

11 |

N |

20 |

T number is the number at the bottom of the T shape

T Number= blue number

To calculate T total add all the numbers inside the T together.

T Total = 1+2+3+11+20 = 37

I will represent T number as T and T total as N

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 |

The numbers in red represent all the places were the T shape cant fit. I will ignore these and only use the squares were the t shape fully fits.

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 |

First I put the T shape onto my 9x9 grid and translated it right by 1 space each time. As shown above I started on 20 and finished on 25 I then constructed the tale below.

T-Number (T) | T-Total (N) | Difference |

20 | 1+2+3+11+20=37 | - |

21 | 2+3+4+12+21=42 | 5 |

22 | 3+4+5+13+22=47 | 5 |

23 | 4+5+6+14+23=52 | 5 |

24 | 5+6+7+15+24=57 | 5 |

25 | 6+7+8+16+25=62 | 5 |

The table above shows the difference between the consecutive T-Totals as the T-Number increases by one. On the above 9x9 grid, the T-Shapes can be seen being translated across the 9 x 9 grid by one square each time. There is a pattern between the T-Totals as the T-Shape is translated each time, as each time the T-Total increases by 5, as shown in the table above.

Middle

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

T-total = (N-21)+ (N-20) + (N-19) + (N-10)+( N)

I then can simplify this to:

T-total = 5N-70

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 |

T-total = (N-23)+(N-22)+(N-21)+(N-11)+(N)

I then can simplify this to:

T-total = 5N-77

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 |

T-total = (N-25)+(N-24)+(N-23)+(N-12)+(N)

Conclusion

I then used simple logic to work out the formula for a 270˚rotation. If 90˚ was 5N + 7 and the angle is 90˚ in the other direction (270˚ of a complete 360˚ rotation) then the formula must be 5N-7. I tested and tried this formula again on a 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 |

The T-Total for this shape is 42 when calculated. When put into the formula it becomes 5(7) + 7 which also equals 42. This proves my theory that:

As with the last formula, I used simple logic to solve the 180˚ rotation formula. If the formula for 0˚ is 5N -7G then the logical formula to solve the 180˚ rotation would be 5N + 7G. I tried and tested the theory below:

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 |

The calculated T-Total for this T-Shape is 57 and the T-Total when put into the formula above also equals 57. This proves that

Conclusion

In this investigation I have successfully established a relationship between T-Total and Grid size, and found a rule for grids ranging from 9-14. I have also found a general rule which will find the T-total for any grid size. I then move onto rotations I looked at 90˚, 180˚, 270˚ and 360˚, I worked out my formulas for rotating the T-shape on a 6x6 grid and successfully found and tested and proved my formula correct. Overall I think the investigation went well.

Evaluation

If I was to do the investigation again I would do grid rotations on larger grid sizes and make a general rule for rotating the T- shape.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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