# I am investigating the relationship between the T-total and the T-number.

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Introduction

Dan Bloom

Introduction

I am investigating the relationship between the T-total and the T-number.

I am going to move the T until I find a pattern.

Collect the results

I started with 22.

The T-Number is 22.

The T-Total is 47 (3 + 4 + 5 + 13 + 22).

I then moved the T-Number along 1.

The T-Number is 23.

The T-Total is 52 (4 + 5 + 6 + 14 + 23).

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 |

I then moved the T-Number along to 24.

The T-Number is 24.

The T-Total is 57 (5 + 6 + 7 + 15 + 24).

I then moved the T-Number along to 25.

The T-Number is 25.

The T-Total is 62 (6 + 7 + 8 + 16 + 25).

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 |

This is my table of my results:

T- Number | T-Total |

22 | 47 |

23 | 52 |

24 | 57 |

25 | 62 |

Patterns

I have noticed that the T- Number’s and the T-Total’s are linked. To get from 47 to 52 to 57 to 62 in the T-Total column, I have to plus 5.

Middle

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72

If we add up the numbers in the T, it should add up to 67.

7 + 8 + 9 + 17 +26 = 67

This shows that the equation works.

Also there is another way of finding out if the equation works.

If you call the T-Total ‘N’ then the number above that, would be N – 9 and so on. I will show this on a diagram:

N – 19 N – 18 N – 17

N – 9

N

If you then put the numbers from the T, (N-19) + (N-18) + (N-17) + (N-9) + (N) = 5N – 63.

This is another way of showing that the equation works.

I will now do the same only this time I will change my grid size from 9 to 10 across. This is to see if there is a pattern.

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 |

Conclusion

I will now change the grid from 10 to 11. Everything else will stay the same.

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 |

T-Number = 24

T-Total = 43

T-Number = 25

T-Total = 48

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 |

T- Number = 26

T- Total = 53

24 | 48 |

25 | 53 |

26 | 58 |

27 | 55 |

T- Number = 27

T- Total = 58

This is my table of results:

T- Number | T-Total |

Because the T-Total is going up in fives I know that it is T = 5N.

5 x 24 = 120

This does not fit so if you take 72 away from 120. That fits.

Therefore the equation must be: T= 5N -72

To prove this equation works, I will use another way to show it.

N - 21 N - 22 N -23

N - 11

N

If you then put the numbers from the T, (N-11) + (N-22) + (N-23) + (N-21) + (N) = 5N – 72.

This is another way of showing that the equation works.

I will now do the same, expect the grid size will be 12 instead of 11.

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

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