# My investigation is to see the relationship between the T-Total and the T-Number of T-Shapes and to find a rule for it so that the T-Total can be easily found.

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Introduction

Sabrina Ul-Hasan 10R

## Introduction

My investigation is to see the relationship between the T-Total and the T-Number of T-Shapes and to find a rule for it so that the T-Total can be easily found.

First I am going to draw a 9 by 9 grid and pick T-Shapes out for example the T-Number would be the number witch is at the bottom of the T-Shape.

e.g.

N -19 1 | N -18 2 | N-17 3 |

N-9 11 | ||

N 20 |

The T-Total would be the number you would get by adding all the figures inside the T-Shape together.

e.g.

N-19 1 | N-18 2 | N-17 3 |

N-9 11 | ||

N 20 |

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

TN=1 + 2 + 3 + 11 +20 = 37

Therefore the T-Total = 37

Here is the grid and my working out for it: -

## A 9 by 9 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 |

N-19 2 | N-18 3 | N-17 4 | ||

N-9 12 | ||||

N 21 |

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

TN= 2 + 3 + 4 +12 + 21 =

T21 = 42

N-19 3 | N-18 4 | N-17 5 |

N-9 13 | ||

N 22 |

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

3 + 4 + 5 + 13 + 22 =

T22=47

N-19 4 | N-18 5 | N-17 6 |

N-9 14 | ||

N 23 |

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

4 + 5 + 6 + 14+ 23=

T23=52

N-19 5 | N-18 6 | N-17 7 |

N-9 15 | ||

N 24 |

Middle

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

N-17 1 | N-16 2 | N-15 3 |

N-8 10 | ||

N 18 |

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

1 + 2 +N- 3 +10 +18 =

T18=34

N-17 2 | N-16 3 | N-15 4 |

N-8 11 | ||

N 19 |

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

2 + 3 + 4 + 11 + 19=

#### T19=39

N-17 3 | N-16 4 | N-15 5 |

N-8 12 | ||

N 20 |

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

3 + 4 + 5 +12 + 20=

#### T20=44

Here is the results table for the 8 by 8 grid :-

T-Number | T-Total |

18 | 34 |

19 | 39 |

20 | 44 |

The T-Total increase’s by 5 so the rule for the 8 by 8 grid would be found in the similar way to the 9 by 9 grid.

To work out the rule for this grid I multiplied 5 by the T-Number and took away the T-Total.

e.g.

1 | 2 | 3 |

10 | ||

18 |

5 x 18 = 90

90 – 34 = 56

As a result the rule for the 8 by 8 grid is :-

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

(By adding together all the n’s you get 5n)

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

(By doing the equation you get left with –56)

5n – 56

Now I am going to do a 10 by 10 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 | 98 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

N-21 1 | N-20 2 | N-19 3 |

N-10 12 | ||

N 22 |

The working out for the rule

TN= N+(N-10)+(N-20)+(N-21)+(N-19)

1 + 2 + 3 + 12 + 22 =

#### T22=40

22 x 5 = 110

110 – 40 = 70

Conclusion

Results table

Grid size | Rule |

8 by 8 | 5n -56 |

9 by 9 | 5n – 63 |

10 by 10 | 5n – 70 |

11 by 11 | 5n – 77 |

g by g | 5n – 7g |

As you can see from the results the rule increases by 7 in each grid. Also the last figure is obtained by multiplying the grid size by 7.

For example:-

The rule for the 8 by 8 grid is 5n – 56

7 x 8 = 56

###### Therefore to show this in algebraic form it would be:

7g

N - (g x 2) +1 | N-(g x 2) | N - (g x 2) -1 |

N-g | ||

N |

Were as the algebraic equation for all of it would be:

5n – 7g

Now I am going to test this rule to see if it works on any grid size. I will do this on a 12 by 12 grid.

12 by 12 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 | 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 |

N-25 1 | N-24 2 | N-23 3 |

N-12 14 | ||

N 26 | ||

5t – 7g is my rule here is the working out

5 x 26 =130

7 x 12 =84

Therefore I think the rule for the 12 by 12 grid will be :

130 – 84

N-25 1 | N-24 2 | N-23 3 |

N-12 14 | ||

N 26 | ||

Now I am going to see if I am right

TN=N+(N-12)+(N-24)+(N-23)+(N-25)

TN= 1+ 2+ 3 + 14 + 26=46

5 x 26 = 130

130 – 46 = 84

Therefore this proves that the rule works on any size grid.

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

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