# T-Total Coursework

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Introduction

Kamran Adnan (10 S2) Maths Coursework T-Total

Part 1

Here we have a grid with nine numbers in a row starting from 1 and ending with 81. There you can see the shape, which is marked in red colour. This shape is called a T-Shape. The T-Shape is always a shape of five numbers.

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 |

The number at the bottom is 20. This number is called the T-Number. The sum of all the five numbers is called the T-Total. In this case the T-Number would be:

1 + 2 + 3 + 11 + 20 = 37

To find out a relationship between the T-Number and the T-Total we need more results in the same 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 |

In this grid we have six more T-Shapes. Now we need to find out their T-Totals.

1 + 2 + 3 + 11 + 20 = 37

2 + 3 + 4 + 12 + 21 = 42

3 + 4 + 5 + 13 + 22 = 47

4 + 5 + 6 + 14 + 23 = 52

Looking at the results we can see that if the T-Number goes one up the T-Total goes five up. This is because by raising the T-Number by one, each number of the T-Shape will go one up. Because there are five numbers in a T-Shape the T-Total goes 5*1 = 5 steps up.

The formula for finding out the T-Total by using the T-Number is 5*N – 63. The following is an explanation how I found this formula out.

I already knew the T-Total for the T-Number 20. As we can see in the results is that every time the T-Number goes up one, the T-Total goes up five. So I had to multiply 20 by 5 because the T-Total goes five up if the T-Number goes one up.

Middle

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

1 + 2 + 3 + 13 + 24 = 43

2 + 3 + 4 + 14 + 25 = 48

3 + 4 + 5 + 15 + 26 = 53

4 + 5 + 6 + 16 + 27 = 58

We can see again that if the T-Number goes one up the T-Total goes five up. This is because by raising the T-Number by one, each number of the T-Shape will go one up. Because there are five numbers in a T-Shape the T-Total goes 5*1 = 5 steps up.

This means again that the formula will have to start with 5*T-Number and end with a subtraction of a number that we will find now.

To find this subtrahend we have to methods. I will start with adding the differences between the T-Number and the four other numbers in the T-Shape (T-Number is 24):

(24 – 13) + (24 – 3) + (24 – 2) + (24 – 1)

= 11 + 21 + 22 + 23

= 77

The second method would be as followed (T-Number is 24):

5*24 – 43 = 77

I got the number 43 by adding the numbers of the T-Shape with the T-Number 24. For this I just looked up to the results.

Now I am going to test this formula.

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 |

The T-Number 98

98*5-77 = 413

Check: 75 + 76 + 77 + 87 + 98 = 413

The T-Number 32

32*5-77 = 83

Check: 9 + 10 + 11 + 21 + 32 = 38

The T-Number 70

70*5 – 77 = 273

Check: 47 + 48 + 49 + 59 + 70 = 273

The T-Number 61

61*5 - 77 = 228

Check: 38 + 39 + 40 + 50 + 61 = 228

Next I am going to use one more grid, which will be smaller than the first one. I will use a grid with only 4 numbers in a row.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

I will start again with the T-Shape that has the smallest T-Number. In this grid it will be the T-Number 10. The T-Total for this T-Shape will be:

1 + 2 + 3 + 6 + 10 = 22

The formula for this shape will again start with 5*T-Number because the T-Shape still has five numbers and if each number goes one up the T-Total will have to go five up. The last number I will again find out by calculating the followings:

5*10 = 50

50 – 22 = 28

Therefore the formula has to be 5*T-Number – 28.

To check whether this formula is correct I will need some more results.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

T-Number 11

11*5-28 = 27

Check: 2 + 3 + 4 + 7 + 11 = 27

T-Number 18

18*5-28 = 62

Check: 9 + 10 + 11 + 14 + 18 = 62

After finding out the formula for each of the grids I looked at the formulae. These were:

9 5* N - 63

11 5* N -77

4 5* N -28

By looking at these formulae I saw that always the last number is a factor of 7. By dividing each number by 7 you get the amount of numbers in each line.

63/7 = 9

77/7 = 11

28/7 = 4

Therefore the general formula for finding the T-Total by using the T-Number is:

T = 5*N – (G*7)

E.G. the formula for the T-Total for a grid with 9 columns will be:

T = 5*N – (9*7)

T = 5*N – 63

There you can see that this is the same formula as I wrote and proved for the first grid, which had 9 columns.

PART 3

180° rotated T-Shape

In this section there is change in the size of grid. There is also transformation and combinations of transformations. I need to find out the investigation of the relationship between the T-Total, the T-Number, the grid size and the transformations.

If I turned the T-Shape around 180 degrees it would look like this.

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 |

Once I have done this I realized that if I reverse the T-Shape I should have to reverse something in the formula.

The formula has a multiplication and a subtraction sign in it and that one of these two has to be turned into the negative. It cannot be the multiplication because in this case the T-Total would be smaller than the T-Number, whereas it is not possible to have a smaller T-Total than the T-Number. Therefore I think I have to turn the subtraction sign into an addition sign. After this the formula should be:

5*N – 63 → 5*N + 63

Now I need to check this formula.

The T-Number will be the number at the top of the T-Shape.

The formula would be:

5*2+63 = 73

Check: 2 + 11 + 19 + 20 + 21 = 73

The formula has worked.

A general formula for finding the T-Total in a 90° clockwise rotated shape this would be:

5*N + 7G

This is because the last number occurs by multiplying the grid size by seven.

270° rotated T-Shape

Next I will find out the formula for a 270° clockwise rotated T-Shape.

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 |

We need to add up all the differences between the T-Number and all the other numbers in the T-Shape. This would be:

12 – 11 = 1

12 – 10 = 2

12 – 1 = 11

12 – 19 = -7

1 + 2 + 11 – 7 = 7

The result of adding up all these differences is 7. This means the formula for a T-Shape, which is 270° clockwise rotated, is 5*N – 7

To prove this I am going to check it on the following shape:

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 |

Conclusion

To check that this formula is correct I am going to show this in a grid. Here the T-Shape has been rotated on the point 67:

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 |

Now I am going to use the formula and put in the original numbers.

(N + a – bG) + (b - aG)

= (38 + 2 + 3*9) + (3 – 2*9)

= 40 + 27 + 3 – 18

= 67 – 15

= 52

With this formula we get to the new T-Number in a 90° rotation. Next we need to find and add the formula for a 90° clockwise rotated shape.

We can find this formula easily by doing followings:

N + 2 – 2G | ||

N | N + 1 | N + 2 |

N + 2 + 2G |

We need to add up each formula in this T-Shape above.

N

N + 1

N + 2

N + 2 – 2G

+ N + 2 + 2G

_ .

= 5N + 7

So the final formula for a 90° clockwise rotation is

5(N + a – bG) + (b - aG) + 7

Conclusion

In this project we have found out many ways in which we can solve a problem that we might have with the T-Shape. It can be in various different positions with different grid sizes. We made the calculation easier by creating a main formula, which can change for all the different circumstances.

The followings are all the formulae I found out and proved in this project. They can work for any grid size.

Normal T-Shape 5N - 7G

90° clockwise rotated 5N + 7

180° clockwise rotated 5N + 7G

270° clockwise rotated 5N - 7G

Translation of T-Shape (N + x – yG)*5 – 7G

90° rotation 5(N + a – bG) + (b - aG) + 7

Page

T = T-Total; N = T-Number; G = Grid size;

x = horizontal movement; y = vertical movement (for translation)

a = horizontal movement; y = vertical movement (for rotation)

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

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