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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).

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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.

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    T number

T total

Increase

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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

...read more.

Middle

New T total

Increase

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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.

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I fist tried reflections in the y-axis.

I got these results

T total

New total

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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

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65

25

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75

35

...read more.

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

...read more.

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