# Investigate the relationship between the T-total and the T-number in the 9 by 9 grid.

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Introduction

T-Totals Coursework

Harriet Blair, 10J

Part 1

Investigate the relationship between the T-total and the T-number in the 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 |

To start with, I took a range of sample Ts to decide whether there is any noticeable correlation between the T-total and the T-number. This would help me decide what to do further on.

Blue T-> T-number = 20

T-total = 37 tn/tt = 0.54

Red T-> T-number = 50

T-total = 187 tn/tt = 0.27

Pink T-> T-number = 79

T-total = 332 tn/tt = 0.24

From this small sample, you can already tell that there is no direct correlation between the T-number and the T-total. However, it should be noted that so far with what has been seen, the higher the T-number, the smaller the tn/tt formula is.

I decided that the easiest method of working out what the relationship between the T-number and the T-total was would be to get a general formula and it was quite easy to do this by replacing the T-number with the variable ‘n’. In this way we have a T-shape where the bottom (T) number is ‘n’ and from this we can work out the others.

Middle

n-19 | n-18 | n-17 |

n-9 | ||

n |

To find the T-total for this general T, you must add all the ‘n’s together. If we give the T-total the symbol ‘t’, the formula is:

t = 5n – 63

This can be rearranged for if you have the T-total and want to work out the T-number from it. That formula would be:

n = (t + 63)/5

This formula solves the first part of the task, however there are some restraints that need to be taken into account. The number ‘n’ cannot be absolutely anywhere in the table, as the rest of the T shape would not be able to fit. Therefore, this equation is only valid if the T-number can be found within the second column to the eighth column and within the third row to the ninth row. Any further out and some of the cells of the T would not fit into the grid and the formula would not work.

Part 2

Use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-totals, the T-numbers and the grid size.

To start with, it is obvious that a new general formula will be required to work with grids of different sizes as the last one relied quite heavily on the fact that the grid was 9 by 9.

Conclusion

t = 5n + 7w + 5x -5wy

The final transformation which I wanted to investigate into was enlargements. As before, I decided that it would probably be best to try to work it out using a general algebraic formula rather than working through lots of examples as there is unlikely to be a direct proportion between the T-number and the T-total. In my example I decided to enlarge the T to double the size by multiplying the width of the T by 2. I would keep the same T-number.

n-5w-2 | n-5w-1 | n-5w | n-5w+1 | n-5w+2 | n-5w+3 |

n-4w-2 | n-4w-1 | n-4w | n-4w+1 | n-4w+2 | n-4w+3 |

n-3w | n-3w+1 | ||||

n-2w | n-2w+1 | ||||

n-w | n-w+1 | ||||

n | n+1 |

This T has the general formulas of

t = 20n – 66w + 10

n = (t + 66w – 10)/ 20

w = (20n – t + 10)/ 66

I also saw that with this T-shape, the formula for translations would also change because there are so many more squares in the T. Instead of adding ‘+5x-5wy’ you would now add ‘+20x-20wy’. This is because there are 20 squares in the T-shape rather than the 5 that there were before.

Conclusion

Overall, there is a relationship between the T-number and the T-shape with whatever alterations I have made to it and general formulas can be made using up to five variables. The most important general formulas which I have found have been highlighted in yellow as I feel that these are the most important results from my investigation.

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

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