# The investiagtion betwwen the relationship of the T-number and T-total

Extracts from this document...

Introduction

Part 1. The investigation into the relationship between the T number (N) and the T total (Tt).

The T number is said to be the number at the bottom of the shape when it is in this shape. The T number always stays as the same box even if rotated.

The T total is the sum of all the 5 boxes.

T I firstly want to find the relationship between the T number and the T total in a 10 x 10 grid, and then in any 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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

T number | T total | Increase | |||||||

22 | 40 | ||||||||

23 | 45 | 5 | |||||||

24 | 50 | 5 | |||||||

25 | 55 | 5 | |||||||

26 | 60 | 5 | |||||||

27 | 65 | 5 | |||||||

28 | 70 | 5 |

From these results I worked out that the Nth term was 5Tn - 70.

However this only applied a 10 by 10 grid and so if I wanted a formula that applied to any grid then I would have to make all the parts to my formula dependant upon the grid size. For this I needed some new lettering. I needed letters that were dependant on the grid size so I used M to represent the increase in value in one movement down the grid, and R to represent the increase in value in one movement right in the grid.

N-2M-R N-2M N -2M+R This grid allowed me to work out a formula

Middle

New T total

Increase

1

90

50

2

140

50

3

190

50

4

240

50

5

290

50

From these results I could see that for on movement down the T total increased by 50, in a 10 by 10 grid. I then deduced the formula Tt + 50Y. Y is the number of movements down the y-axis. However unlike the previous formula this one is limited only to a 10 by 10 grid. I therefore had to add a variable that is dependant on the grid size to it. Any movement down will add 5M to the value so therefore Tt + Y (5M) would be the formula that encompasses any grid size.

However I wanted a formula which would apply to any combinations of translations do I had to combine the to formulae. I got Tt + 5X + Y (5M).

I then moved on to reflections this time I aimed to get a formula for any x-axis reflection and any y-axis one.

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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

I fist tried reflections in the y-axis.

I got these results

T total | New total |

40 | 55 |

45 | 60 |

50 | 65 |

55 | 70 |

The difference was constant at 15. I therefore devised the formula Tt + 15N. However this was only for a reflection in the line touching the extreme right edge. I therefore experimented with changing the distance between the edge and reflection line.

Distance | New total | Increase |

0 | 55 | 15 |

1 | 65 | 25 |

2 | 75 | 35 |

Conclusion

NTt = 125 + 285 - 560

NTt = -150

However this is incorrect. It is because the formula for enlargement does not work if the shape has been inverted. Therefore my formula cannot mix x-axis reflections with enlargements.

Limitations of Formulae

I have put below all the limitations of my formulae:

- The formula 5Tn - 7M (T number to Total) only applies when the shape is upright
- The formulae Tt + 5X + Y (5M). (Translation), Tt + 15R + D (10R). (Y-axis reflection) and Tt + 19M + D (10M). (X-axis reflection) only apply to an original sized shape
- The formula 5(S (T + (3M + R) – M (0.5A2+0.5A) + R (0.5A2+0.5A))) – 7M (E3). Can only be used in an upright shape and cannot make the shape smaller than 1 number per square
- NTt = ((T total + first transformation clause) + second transformation clause) _ + third transformation clause) etc. Cannot work for an enlargement after an x-axis reflection and in it enlargements can only be the final transformation.

Notations used

N = T number

Tt = T total

R = Increase in 1 movement right on the grid

M = Increase in 1 movement down the grid

X = Number of translations up the x-axis

Y = Number of translations down the y-axis

D = Distance between edge closest to reflection line and the line

S = Number of numbers in one square of the shape

T = The number at the bottom left of the T number square

A = The enlargement factor -1

E = The enlargement factor

NTt = New T total

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

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