# 1) Investigate the relationship between the T-total and the T-number. 2) Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

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Introduction

T-Totals

Introduction

If you look at the 9x9 grid with the T-shape, you can see that the total of the numbers added together is 37 because it is1+2+3+11+21 which equals 37.

This is what we call the T-total (37)

And T-number is the number at the bottom of the T-shape which in this case is 20

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 |

My tasks

The tasks I have been set are:

1) Investigate the relationship between the T-total and the T-number.

2) Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

3) Use grids of different sizes again. Try other transformation and combinations of transformations. Investigate relationship between the T-total, the T-numbers, the grid size and the transformations.

Standard 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 |

If I place the results from 7 T-Shapes into a table then it would look like this.

T-number | 20 | 21 | 22 | 23 | 24 | 25 | 26 | |

T-total | 37 | 42 | 47 | 52 | 57 | 62 | 67 |

This table clearly shows that the numbers go up by 5 every time the T-number goes up by 1.

Now I can use trial and error to try to find the formula, I will then test what I believe to be the correct formula to see if it is.

Because there is a difference of 5 between the T-totals I think that it is a logical place to start the formula.

Middle

24-1=23

24-2=22

24-3=21

24-13=11

TOTAL=77

I can now take an educated guess that this is the number that changes as the grid size alters. I now have to alter the formula to allow the size of the grid to be multiplied or added to another number to reach the difference of 77 I require. So I did this and I found the formula, with g representing the grid number.

5T-(7g)

I shall now test this formula on another grid size to make sure that it is correct.

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 |

With my formula I have calculated that the T-total for T-number 22 on a 6 by 6 grid will be 5x22-(7x6) =68

After studying the T-shapes for some time now and studying my formula, I have realize that A always equals T - 2 x The grid size (g) – 1, B equals T - 2g, C = T – 2g + 1, and D = T – g

As the T-Total is A+B+C+D+T I can conclude therefore that the

T-total = T – 2g – 1 + T – 2g + T – 2g + 1 + T – g + T

This can be simplified to 5T – 7g my prior formula.

I have now proven that my formula works on smaller sized grids.

Conclusion

Remainder- this is the column of the T-number. We know this is correct because 9-2= 7 so the middle seven columns are valid and two is in the centre.

So the three steps to validating a T-number are:

1. T/S - The T-Number (T) is divided by the Grid Size (S) , the answer will be a number and a remainder (R=Row, C=Column).

T / S = R .C

2. Row - The row number +1 must be higher then 2 and lower then the grid size +1

(The inequality 2 > (R+1) < S+1)

3. Column - The column number must be lower then one and also lower then the grid size -1

(The inequality 1 > C < S-1)

If the two numbers fit the inequalities then the T-Number will fit inside the grid. I will now test the formula using random figures.

T-Number | Manually | Using Formula |

2 | No | No |

11 | No | No |

20 | Yes | Yes |

28 | No | No |

63 | No | No |

42 | Yes | Yes |

Conclusion

In conclusion I have worked out many different formulae, including how to work out the T-total from just the T-number, how to work out a rotated T’s T-total from the T-number, working the T-total out by using vectors, checking if all of a T-shape is in a grid, and checked everything I have done. I enjoyed this work as it used a lot of formulae and think I explained my method to the best of my abilities.

By Keir Wyndham-Ayres

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

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