# T totals - translations and rotations

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Introduction

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T-totals coursework

By Michael Ogolo

My T number is 20 as you can see on my 9by9 grid and I will be representing this as N in my equation. My T total is 1+2+3+11+20= 37. The number in my T shape directly above the T number is 9 places back in the grid so it is N-9. The number directly above this number is also 9 places back in the grid so it is N-9-9= N-18. The two remaining numbers in the T shape are N-18+1 and N-18-1. Thus the T total is:

N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63

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37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

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55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 78 | 79 | 80 | 81 | 82 |

Rotated translation 90 degrees anti-clockwise

My T number is 21 as you can see on my 9by9 grid and I will be representing this as N in my equation. My T total is 28+19+10+20+21=98. The number in my T shape directly above the T number is 1 place back in the grid so it is N-1. The number directly above this is 1 place back in the grid so it is N-1-1= N-2. The two remaining numbers in the T shape are N-2+9 and N-2-9. Thus the T total is:

N+ (N-1) + (N-2) + (N-2+9) + (N-2-9) = 5N-63

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 | 78 | 79 | 80 | 81 | 82 |

Vertical translation, rotated 180 degrees clockwise

My T number is 11 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 11+12+13+4+22=62. The number in my T shape directly above the T number is 1 place ahead on my grid so it is N+1.

Middle

N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63

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 | 78 | 79 | 80 | 81 | 82 |

Horizontal translation

My T number is 25 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 25+16+7+6+8= 62. The number in my T shape directly above my T number is 9 places back on my grid so it is N-9.The number in my T shape directly above this is also 9 places back on the grid so it is N-9-9= N-18. The two remaining numbers in the T shape are N-18+1 and N-18-1. Thus the T total is:

N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63

T-Number T-Total

20 37

21 42

22 47

23 52

24 57

25 62

Looking at my results above I have spotted a relationship between the T-Totals as I transform the T-shape horizontally. I have noticed that as I transform the T-shape horizontally my T-Total is increased by 5 I will test my theory by making a prediction on my T-Number 26. My prediction is that when my T-Number is 26 my T-Total will be 67.

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64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 78 | 79 | 80 | 81 | 82 |

Horizontal translation

My T number is 26 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 26+17+8+7+9= 67. The number in my T shape directly above my T number is 9 places back on the grid so it is N-9. The number directly above this is also 9 places back on my grid so it is N-9-9= N-18. The remaining two numbers are N-18+1 and N-18-1.

Conclusion

N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42

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My T-number is 16 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 16+10+4+3+5= 38. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid so it is represented as N-6-6= N-12. The two remaining numbers are N-12-1 and N-12+1. Thus my T-total is:

N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42

T-number T-total

14 28

15 33

16 38

Using this data in my table I will try and predict my 17th T-number in my 6by6 grid. Analyzing the data in my table I have picked up on a sequence which is when my T-number increases by 1 my T-total increases by 5.

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25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 |

My T-number is 17 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 17+11+5+4+6= 43. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid so it is represented as N-6-6= N-12. The two remaining numbers in the T-shape are N-12-1 and N-12+1. Thus my T-total is:

N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42

I have proven my prediction correct because my T-total is 43 which is an increase from the 16th T-number which was 38.

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

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