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This Piece of course work is to investigate the relationship between the T Total and the T Number . The T shapes will be 3 squares across and 3 squares down and they will be on a 9*9 grid.

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Introduction

This Piece of course work is to investigate the relationship between the T Total and the T Number . The T shapes will be 3 squares across and 3 squares down and they will be on a 9*9 grid.

Firstly I will draw some T shapes onto the grid and see if I can  spot a relationship.

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T Number=20      T Total=1+2+3+11+20=37

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T Number=33       T Total=14+15+16+24+33=102

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T Number=71      T Total=52+53+54+62+71=292

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T Number=71          T Total+55+56+57+65+74=307

Although those results could give me enough information to find a formula to find the T Total from the T Number, Now I will draw up some more T Shapes but this time all adjacent so that when I draw a table it will be easier to spot the relationship.

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T Number=20       T Total=1+2+3+11+20=37

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T Number=21       T Total=2+3+4+12+21=42

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T Number=22      T Total=3+4+5+13=22=47

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T Number=23       T Total=4+5+6+14+23=52

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T Number=24      T Total=5+6+7+15+24=57

If I now put the these results into a table I will be able to see a pattern more clearly

T Number :  20  21  22  23  24

…………..……………………………..  

T Total      :  37  42  47  52  57          

The T Total goes up in 5’s because there are 5 squares in each T and each Square in The T goes up 1 adding up to 5.

...read more.

Middle

Again I will substitute my T Number with N and then change the other numbers in relation with N so I can find the formula.

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Change this Yellow T To This Red T

N-11
N-10
N-9

N-5


N

If I add up all the N T I end up with the formaula t=5n-35

So to check this with our original answer of just adding up the answer of the formula so

T=5n-35       5n=90

90-35=55

T=7+8=9+13+18

T=55

So that proves that 5n-35 is correct on a 5*5 grid

Now I Will test out a 4*4 grid to make sure that every grid has a different formula.

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To find the formula I will again need to substitute the T number with N and the other numbers in relation with N

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change this blue T into this Green T

N-9
N-8
N-7

N-4


N

So if you add up all the N T Shape we get the formula T=5N-28

So Now if I add up the original T shape I get:

T=6+7+8+11+15

T=47

Now I need to test if the formula gives me the same answer

T=5n-28          5n=75

T=75-28

T=47   So that proves that 5n-28 is the correct formula for a 4*4 grid.

...read more.

Conclusion



90 Clockwise
T=5n+7

180

T=5n+7g

90 Anti clockwise
T=5n-7

Note that 90 Degrees Rotations do not include ‘g’ the grid size so these formulae are independent of grid size.

I will now Rotate a T shape 180 Degrees about an external point using the vectors          {2}

{-1}

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I have noticed that if u double the vector u reach the corresponding T Number straight away and if we use x and y instead of numbers we will now have 2*{ x  }

                                                                                        {-yg}

So if we go 2x across from The T Number 30 we get to 34 and go down 2*yg

we take away two grid sizes which is equal to 18 which is also equal to going down 2 squares so we get the formula:

T=5N+7G+5(2x-2YG)

I will now apply a combined transformation so all I need to do is find the formula for the second transformation as I have the formula for a 180 degrees rotation about an external point: T=5N+7G+5(2A-2BG)

So If I Now translate by a vector {s}

                                            {t} the formula will also include the translation                       {s}

{-gt} for each T Number , therefore the total; formula for the T-Total will be:

T=5n+7g+5(2x-2YG+S-GT)

Where T=T Total

          N= T Number

          G=Grid Size

          {x} ……… is vector for point of rotation

          {y}

          {s}…………..is Translation Vector

          {t}

...read more.

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