# This project consists of the investigation between patterns between T-numbers and T-totals.

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Introduction

- Investigation

## Introduction

This project consists of the investigation between patterns between T-numbers and T-totals. My investigation consists of different aims, but all of these aims have to do with this mathematical “T”. I believe that there is a direct link between these two, as the T-total increases depending on the T-number. I think that there is nth sequence relating the T-number and the T-total, and therefore I will try and investigate this.

First, I will use a 9 by 11 grid and find the relationship between the T-number, and the T-total in this grid. I will draw other girds and also find out the correlation between the translation of T-numbers, T-totals and the gird size.

After this, I will go back to my initial 9 by 11 grid, and rotate the “T” and try to find the possible relationship of T-numbers and T-total of the rotated “T”. I believe that there is a formula for the T-total of T-shapes in any grid and in any rotation.

Subsequent to this investigation, I will use and enlargement of my initial “T” and use different examples to show the relationship between the enlargement and the T-number and T-total. From all my different investigations I will draw my conclusions giving explications for these.

- Method: Task I.

1. I will draw a 9 by 11 grid,

Middle

i.e.

79 | 80 | 81 |

89 | ||

98 | ||

98-19 | 98-18 | 98-17 |

98-9 | ||

98 |

i.e

48-19 | 48-18 | 48-17 |

48-9 | ||

48 | ||

29 | 30 | 31 |

39 | ||

48 |

As it worked for all the T’s, I put the formula together creating a simple formula, which is used in form of substitution for achieving the T-total from the T-number.

- Method: Task II

- In this task I will different size grids, to investigate the correlation between the T-number, T-total and the translation of the T-shape in any grid size.
- First I will use a grid size 12 by 12, creating ten T’s for this investigation. I will do the same with a 5x5 grid. I will try to see the algebraic formula as found for the 9 by 11 grid.
- With the different T-totals and T-numbers I will create a result table to be able to see any observations.
- I will right down any observations noticed.
- I will go deeper into the investigation to try to find any algebra correlation between T-total and T-numbers of different grid sizes.

The 12 by 12 grid, and the ten T-shapes:

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 |

Results=

T-number | T-total | T-number | T-total | ||

26 | 46 | 76 | 296 | ||

40 | 116 | 82 | 326 | ||

54 | 186 | 123 | 531 | ||

62 | 226 | 126 | 546 | ||

68 | 256 | 129 | 561 |

The 5 by 5 grid and the “T’s”:

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 |

Results=

T-number | T-total | |

12 | 25 | |

18 | 55 | |

19 | 60 | |

22 | 75 |

- Observations

From the different tables of result, we can make the following statements:

- In the 12 by 12 grid the increase of 1 in the T-number means also an increase of five.
- This also works for the 5 by 5 grid.
- The odd T-numbers have a T-total that ends in 1, and in the even T-numbers the T-total ends in a 6. Which means that there is a sum of 5 each time. 1 + 5 = 6, 6 + 5 = 11
- This also works for the 5 by 5 grid. But the odd numbers finish in 0, and the even T-numbers end in 5. 0 + 5 = 5, 5 + 5 = 10

- Representing my different observations.

Before I had found a equation for the T-total for the 9 by 9, but this formula doesn’t work for the rest of the grids.

i.e. 5n – 63 = T-total

98 | 99 | 100 |

110 | 111 | 112 |

122 | 123 | 124 |

123 x 5 – 63 = 552

T-number= 123, T-total = 531

As this formula didn’t work I found another formula for the 12 by 12 grid, the same way I investigated the formula for the 9x9 grid.

n-25 | n-24 | n-23 |

n-12 | ||

n |

This formula isn’t valid for the size five grid, as the numbers after the 5n, change, the formula for a size grid five would be:

n-11 | n-10 | n-9 |

n- 5 | ||

n |

As I said before in both grids 12x12 and 5x5 have an increase of five in the T-total when the T-number is increased by one, and this also works with the 9 by 9 grid. So we can say that the rule n+1= t + 5 is true for all grids, where n is the T-number, and t is the T-total. The difference between the outcomes is notices by the starting number from where to add the 5 to. These are the lowest incomes for “T’s”

i.e. In 5x5 grid.

T-number= 12, T-total= 25

T-number= 13, T-total= 25 + 5 = 30.

The starting number to add, is 2

In 12x12 grid.

T-number= 26, T-total= 46

T-number= 27, T-total= 46 + 5 = 51

The starting number to add is 1.

In 9x11 grid.

T-number= 20, T-total= 37

T-number= 21, T-total= 37 + 5 = 42

The starting number to add is 1.

I believe, that there is an equation to link the different translations and I managed to find an equation for any translation, horizontal and/or vertical, that includes the “magic” number five. The result of this equation gives the T-total of the translated shape in this way:

Where v is the middle number of the T and g is the grid size. “a” and “b” are the translations. “a” is the vertical translation and “b” is the horizontal translation.

To prove this equation is correct I use a grid size of 10 , and an initial T, which was translated +2 vertically, and -2 horizontally.

41 | 42 | 43 |

51 | 52 | 53 |

61 | 62 | 63 |

Initial T:

V= 52

Translation: Translated T:

A= 2

23 | 24 | 25 |

33 | 34 | 35 |

43 | 44 | 45 |

T-total= 150

B= 1

With formula:

=(5x52-2x10)-(2(5x10)-5 x-2

= 240 – 100 + 10

= 150

This proves my equation is correct as; the T-total of the translated shape is equal to the mathematically translated shape, using the equation. I can state;

Any combination of translations (vertically and horizontally), can be found by using the equation

Were v is the middle number of the initial T, g is the grid width, a is the number by which to translate vertically and b is the number by which to translate horizontally. It is also apparent that this equation can be used for any singular vertical or horizontal translation, as the other value (vertical or horizontal) will be equal to 0. therefore it will cancel out that part of the equation as it would equal 0.

- Algebraic formula connecting T numbers and totals, with the different grid sizes.

I have made the substitution of any “T” with “n’s” (T-numbers) and “G’s”, the grid size.

n-2G-1 | n-2G | n-2G+1 |

n-G | ||

n |

In total this is= n-2G-1 + n-26 + n-2G+1 + n-G + n

= 5n - 7G

I have to check if this works for all grids.

- 5 x 5 grid.

Conclusion

1 | 2 | 3 | 4 | 5 |

10 | 11 | 12 | 13 | 14 |

19 | 20 | 21 | 22 | 23 |

T-number= 21

T-total= 48

Stretch= 2

The T-number is increased by one so:

2 | 3 | 4 | 5 | 6 |

11 | 12 | 13 | 14 | 15 |

20 | 21 | 22 | 23 | 24 |

21 + 1 = 48 + 5 + 2

= 55

From my investigations I can then state that the formula

Is applicable for all T-translations that are increased by one in any grid size with any even stretch.

- Conclusion

Throughout this investigation I have found different relations between T-numbers and T-total in a T-shape. There is a series of formula’s that I have proved worked, that I now state:

- With a 9 by 11 grid the equation is correct anywhere in the grid.
- Changing the vertical size of this grid makes no change, but if the width of the grid is changed the equation is changed.
- The equation for a 12 size grid is and this works anywhere on this grid size.
- The equation for the five size grid is. This works anywhere in this sort of width.
- The relationship between all the different grid sizes is the 5n, and the number that this is taken away from, as they are all multiples of seven.
- There is two equations relating any translation in any grid size. These are:
- When the T-shape is rotated there are different rules for each individual rotation to find out the T-total. I have investigated these first with a size 9 grid and they are:

= 5n- 63 =5n+ 63

= 5n – 7 = 5n + 7

- I investigated these formula’s even further, and developed them for them to work with a T-shape rotation in any size grid. These formula’s were:

= 5n- 7G =5n+ 7G

= 5n – 7 = 5n + 7

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

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