# T Total Investigation

Extracts from this document...

Introduction

Razvan Alexa 11R10th March 2002

### Math’s Coursework

T-Total

I have been given a task to translate the T-Shape to different positions on the 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 |

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.

- Investigate the relationship between the T-Total and the T-Number.

N |

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

N-19 | N-18 | N-17 |

N-9 | ||

N |

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 |

T. NumberT. Total

20 37

32 97

26 67

56 217

71 292

77 322

1+2+3+11+20=37

13+14+15+23+32=97

7+8+9+17+26=67

37+38+39+47+56=217

52+53+54+62+71=292

58+59+60+68+77=322

## You will notice that the center column of the T-Shape is going up in 9’s because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:

## T-total = T-19+T-18+T-17+T-9+T

T-Total = 5T-63

Now to test this formula to see if it works

For T-total I will use the letter X

For the T-Number I will use the letter T

So X = 5T-63

T = 20

X = 5x20-63

= 100-63

= 37

Middle

T. NumberT. Total

16 31

20 51

44 171

48 191

1+2+3+9+16=31

5+6+7+13+20=51

29+30+31+37+44=171

33+34+35+41+48=191

1 | 2 | 3 |

9 | ||

16 |

## X=5x16-49 (1+2+3+9+16=31)

X=80-49

X=31

I will now try it for two more of my t-shapes to make sure the formula works correctly.

5 | 6 | 7 |

13 | ||

20 |

## X=5x20-49 (5+6+7+13+20=51)

X=100-49

X=51

33 | 34 | 35 |

41 | ||

48 |

## X=5x48-49 (33+34+35+41+48=191)

X=240-49

X=191

I will now try the formula on a 6 by 6 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 |

The formula is X=5T-7G

Which changes to X=5T-7x6

X=5T-42

T. NumberT. Total

14 28

23 73

33 123

1+2+3+8+14=28

10+11+12+17+23=73

20+21+22+27+33=123

1 | 2 | 3 |

8 | ||

14 |

## X=5x14-42 (1+2+3+8+14=28)

X=70-42

X=28

I will now try it for two more of my t-shapes to make sure the formula works correctly.

10 | 11 | 12 |

17 | ||

23 |

## X=5x23-42 (10+11+12+17+23=73)

X=115-42

X=73

20 | 21 | 22 |

27 | ||

33 |

## X=5x33-42 (20+21+22+27+33=123)

X=165-42

X=123

I have tried this formula out on three different grid sizes and the formula works, the formula is: X=5T-7G, when Grid Width (= G), the T-Number (= T), and the T-Total (= X)

3) Grids of different size using other transformations and combinations of transformations to Investigate relationships between the T-Total, the T-Numbers, the Grid sizes and the transformations.

Below is a grid with parts of the formula in. I will use this in a minute to figure out a formula.

1 | 2 | 3 | . | G |

G+1 | G+2 | G+3 | . | 2G |

2G+1 | 2G+2 | 2G+3 | . | 3G |

For a “ T ” in the grid

1 | 2 | 3 | 4 | 5 | ... | G |

G+1 | G+2 | G+3 | G+4 | G+5 | ... | 2G |

2G+1 | 2G+2 | 2G+3 | 2G+4 | 2G+5 | ... | 3G |

3G+1 | 3G+2 | 3G+3 | 3G+4 | 3G+5 | ... | 4G |

4G+1 | 4G+2 | 4G+3 | 4G+4 | 4G+5 | ... | 5G |

2 squares right

1 square down

From 2G+2 to 3G+4

This looks like adding numbers of squares down to the number of G’s and the number of squares right to the other number. If we start with a T-number (T) then moving 4 right and 3 down

The new T-number is:

T+3G+4

We can use “h” for the number of places right and “Y” for the number of places down.

We could write this as (h/y), a column vector:

Now X+(4/3)=X+3G+4

And X+(h/y)=X+yG+h

For all of the “ T’s ” the T-number is the same as shown below:

The other “ T’s ” that I can work out the formula for are shown bellow.

A 9 by 9 grid can be used for the other 3.

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 |

T. NumberT. Total

- 142

41 198

41 212

41 268

22+23+24+32+41=142

30+39+48+40+41=198

34+43+52+42+41=212

58+59+60+50+41=268

I will now try to find out the formula for my t-shape when the T is 90° from it upright position when (T=T. Total) and (G=grid size).

T+2-G | ||

T | T+1 | T+2 |

T+2+G |

All of the formulas in the grid above added together gives me an overall formula of:

T+(T+1)+(T+2-G)+(T+2)+(T+2+G)

5T +7

Now that I know the formula for the T-Shape 90° from it upright position, I will now test it 3 times, with different sized grids.

I will test my new formula, in an 8 by 8 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 |

Conclusion

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 |

7G+5T

=7x8+5x29 (T. Total: 44+45+46+37+29=201)

=56 + 145

=201

A 7 by 7 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 |

7G+5T

=7x7 + 5x18 (T. Total: 31+32+33+25+18=139)

= 49 + 90

= 139

Also, now on a 6 by 6 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 |

7G+5T

=7x6 + 5x15 (T. Total: 26+27+28+21+15=117)

= 42 + 75

= 117

I will now try to find out the formula for my t-shape when the T is 260° from it upright position when (T=T. Total) and (G=grid size).

T-2-G | ||

T-2 | T-1 | T |

T-2+G |

All of the formulas in the grid above added together gives me an overall formula of:

T+(T-1)+(T-2+G)+(T-2)+(T-2-G)

5T-7

Now that I know the formula for the T-Shape 260° from it upright position, I will now test it 3 times, with different sized grids.

I will test my new formula, firstly on a 9 by 9 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 |

5T-7

=5x30 – 7 (T. Total: 19+28+37+29+30=143)

=150 – 7

=143

I will now try it on an 8 by 8 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 |

5T – 7

=5x29 – 7 (T. Total: 19+27+35+28+29=138)

=145 – 7

=138

I will now try it on a 7 by 7 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 |

5T – 7

=5x40 – 7 (T. Total: 31+38+45+39+40= 193)

=200 – 7

=193

Whilst doing this investigation I found out that for the 2 T’s which were sideways like this:

Had the formula’s 5T – 7 and 5T + 7 which did not have a G (Grid Size) in the formula, because they would cancel each other out thus telling me that this formula was not affected by the grid size in order for me to find the T. Total with my formula’s.

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

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