# T-Total and the T-Number.

Extracts from this document...

Introduction

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Aim:

My aim is to investigate the relationship between the T-Total and the T-Number. I will first do so using a 9x9 grid. After that, I shall use grids of different sizes. Using algebra, I will find the formula for finding the T-Total using the T-Number for all grid sizes. For further investigation, I plan to translate the "T" into different directions. I plan to put all my formulas in a "Formula Table" at the end of my investigation.

9x9:

T-Number | 20 | 23 | 31 | 44 | 57 | 62 |

T-Total | 37 | 52 | 92 | 157 | 222 | 247 |

This is a 9x9 grid with 6 "T" shapes inside it. Some of the "T"s are overlapping but this does not really matter. Next to it is a table showing each of the "T"s properties, (T-Number & T-Total). The first pattern being spotted is obviously that as the T-Number increases so does the T-Total. This meaning that the T-Total is directly proportional to the T-Number. I now need to figure out a formula for finding the T-Total using T-Number in a 9x9 grid, which I've done below showing the routes of my formula.

n-19 + n-18 + n-17 + n-9 +n = 5n - 63 = T-Total

Above is a "T" representing itself in a 9x9 grid, and next to it as you can see is the formula for working out the T-Total knowing the T-Number.

Middle

121

171

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43 | 44 | 45 | 46 | 47 | 48 | 49 |

This is a 7x7 grid with 6 "T" shapes inside it. Again they're overlapping which does not matter. Next to the grid is a "T" property table. I now have to figure out a formula for working out any number in the "T" knowing the T-Number in a 7x7 grid. The formula for working this out is shown below.

n-15 + n-14 + n-13 + n-7 +n = 5n – 49 = T-Total

Thisis a "T" representing itself in a 7x7 grid, and next to it as you can see is the formula for working out the T-Total knowing the T-Number.

I am now going to test the formula I have found.

234(5 x 17) -49 = 85-49 = 36 = T-Total FORMULA WORKS

10

17

101112(5 x 25) -49 = 125-49 = 76 = T-Total FORMULA WORKS

18

25

192021(5 x 34) -49 = 170-49 = 121 = T-Total FORMULA WORKS

27

34

By now, I have found 4 formulas for working out any numbers in 7x7, 8x8 and 9x9 grids, knowing the T-Number. Another pattern I have now spotted in the formulas being discovered by myself is that as the grid size decreases by a size the formulas decreases by 7. Basically the formula is 5n takeaway the grid size multiplied by 7.

9 x9 Grid | 8 x 8 Grid | 7 x 7 Grid |

n-19 + n-18 + n-17 + n-9 +n = 5n - 63 | n-17 + n-16 + n-15 + n-8 +n = 5n – 56 | n-15 + n-14 + n-13 + n-7 +n = 5n – 49 |

5n –(9x7) = 5n - 63

5n –(8x7) = 5n – 56

5n – (7x7) = 5n - 49

Now that I know that the formula goes up and down in 7s, I am going to attempt to predict the formula for a 10 x 10 and a 5 x 5 grid.

10x10:

Knowing the pattern for the formulas, I predict that the formula for a 10 x 10 grid will be 5n – 63 plus 7. Therefore it will be 5n – 70, because 63 + 7 = 70 and because 10 x 7 = 70. I will know check my prediction. Below is a part of a 10 x 10 grid.

T-Number | 22 | 26 | 28 | 32 |

T-Total | 40 | 60 | 70 | 90 |

Above is part of a 10 x 10 grid and next to it is a table showing the properties of the 4 "T"s present in the part of the grid. Below is the working out for the formula, so then I can check if my predictions were correct.

n-21 + n-20 + n-22 + n-10 +n = 5n – 70 = T-Total

The formula I have found matches the one I predicted it would be. But that's not it; I now have to check if the formula is correct and works positively.

Prediction check:

(5 x 22) -70 = 110-70 = 40 = T-TotalPREDICTION CORRECT

(5 x 26) -70 = 130-70 = 60 = T-TotalPREDICTION CORRECT

(5 x 32) -70 = 160-70 = 90 = T-TotalPREDICTION CORRECT

(5 x 28) -70 = 140-70 = 70 = T-TotalPREDICTION CORRECT

As you can see, my predicted formula totally matched the one I found by investigating the 10 x 10 grid, and positively worked. Just to make sure, I am now going to do the same for a 5 x 5 grid as mentioned earlier on.

5x5:

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 |

T-Number | 12 | 13 | 17 | 24 |

T-Total | 25 | 30 | 50 | 85 |

Conclusion

Conclusion:

I can conclude that in the time available to me, two weeks precisely, I think I have succeeded in investigating the relationship between the T-Total and the T-Number. I have used grids of different sizes and translated the "T"s to different directions and positions on the grids. By doing so I have found a number of formulas for working out any number including the T-Total knowing the T-Number in a number of grids.

Evaluation:

I would say that this investigation has been a success. I managed to find a number of rules and formulas to work out numbers in a "T". I do believe that I followed the guidelines given pretty well.

Unfortunately, my investigation was hindered by 1 thing – this was my lack of time to carry out the investigation as far as possible ( i.e. use more grids and test each of the formulas present in my formula page a few more times).

If I were to redo this investigation, I would make sure that I set aside enough time to do a proper job of it.

Wais Naeemi 10E Y-Band

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

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