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Relationships with the T-shapes, the T-total and the position of the stair shape on the grid.

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Introduction

T- Total

By Gary Holmes

Introduction:

                In this investigation in which I am presenting, I am going to look at the relationships with the T-shapes, the T-total and the position of the stair shape on the grid. I will be observing patterns and I will look at the other stairs to see if I can notice other formulas for these. Finally, I hope to be able to find the formula for any size grids and any T-total

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Look at the T-shape drawn on the 9 by 9     number grid.

The total of the numbers inside the T-shape is 1+2+3+11+20=37

This is called the T-total.

The number at the bottom of the T-shape is called theT-number.

The T-number for this T-shape is 20.

T

If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.

T-17

T-18

T-19

T-9

T

You will notice that the centre column of the T-Shape is going up in 9's because of the table size.

...read more.

Middle

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X=5X33-42 (20+21+22+27+33=123

X=165-42

X=123


Now I have worked the formula out for the T-Shape in the one position, I am going to translate the T-Shape to different positions and investigate the relationships between the T-Total, the T-number and the grid size.

A 3 by 3 grid can be used for the other 3.

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T

T+G

T+2G+1

T+2G

T+2G-1

T

T+G

T+2G

T+2G-1

T+2G+1

7G+5T

X=7G+5T=7x3+5x2

=21+10 (2+5+7+8+9=31)

=31

I will now try this new formula on a 4 by 7 grid.

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7G+5T

x=7x4+5x10 (17+18+19+14+10=78)

=28+50

=78

I will now try it on a 8 by 3 grid.

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7G+5T

x=7x9+5x2 (19+20+21+11+2=73)

=56+10

=73

I have tried this new formula for an inverted T and it works, the formula is: 7G+5T

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T-1-G

T-2

T-1

T

T-2+G

T

T-1

T-2

T-2-G

T-2+G

5T-7

X= 5T-7=5x6-7

=30-7 (1+4+7+5+6=23)

=23

I will now try the new formula on a 4 by 4 grid

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5T-7

X=5x11-7 (5+9+13+10+11=48)

=48

And on a 4 by 9 grid.

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5T-7

x=5x19-7 (13+17+21+18+19=88)

=88

I have tested this new formula for a side ways T the formula works it is: 5T-7

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T+2-G

T

T+1

T+2

T+2+G

T

T+1

T+2

T+2+G

T+2-G

5t+7

The formula is 5T+7

X=5T+7=5x4+7

=20+7 (3+6+9+5+4=27)

=27

I will now try the new formula on a 6 by 6 grid.

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...read more.

Conclusion

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T- Number = 12

Purple T-Total = 25

Grey T-Total = 67 a difference of 42….. The formula 5T - 7G + 42

Now I'm going to see what happens on a 6 by 6 width Grid

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T- Number = 14

Purple T-Total = 33

Grey T-Total = 82 a difference of 49….. The formula 5T - 7G + 49

7 by 7 width Grid

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T- Number = 16

Purple T-Total = 41

Grey T-Total = 97 a difference of 56….. The formula 5T - 7G + 56

Table of Results

Grid width

Purple T-Total

Grey T-Total

Difference between the two T-Total

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25

67

42

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33

82

49 Goes up in 7's

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97

56

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Now I will rotate it again 180° from it's original place

T- Number = 12

Purple T-Total = 25

Grey T-Total = 95 a difference of 70….. The formula 5T - 7G + 70

Now I'm going to see what happens on a 6 by 6 width Grid

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T- Number = 14

Purple T = 28

Grey T = 112 a difference of 84….. The formula 5T - 7G + 84

7 by 7 Grid

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T- Number = 16

Purple T-Total = 31

Grey T-Total = 129 a difference of 98….The formula 5T - 7G + 98

Conclusion: The amount of numbers in the T decide the number before N in my formula. The number before the W is found by
(T-total - 2T) divided by the top right number of the grid.

By Gary Holmes

...read more.

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