# T -Totals. From this 9*9 grid I will collect a number of T Shapes with T Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

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Introduction

Sohaib Mohammad 11H -- Maths Coursework -- T Totals

T –Totals

For this coursework I have been given a task which is to investigate the relationship between the T-total and the T-number. To start my investigation I will do this by working on a 9*9 grid. From this 9*9 grid I will collect a number of T –Shapes with T –Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

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 |

Definition of the T –Total and T –Numbers

Consider the numbers in the T –Shaped figure shown in the grid by thick lines, redrawn underneath separately for clarity. The integer at the bottom of the T –Shape is 20 this I known as the T –number for the T –Shape:

1 | 2 | 3 |

11 | ||

20 |

The T –Total is defined as the sum of all the integers shown in the above T –Shaped figure. For example the T –Total for the T –Shape above = 1+2+3+11+20 = 37

I firstly begin the investigation by starting of with a T shape, the following shows 4 T –Shapes collected from the first row:

2 | 3 | 4 |

12 | ||

21 |

T –Number = 21

T –Shape = 2+3+4+12+21 = T –Total = 42

3 | 4 | 5 |

13 | ||

22 |

T –Number = 22

T –Shape = 3+4+5+13+22 = T –Total = 47

4 | 5 | 6 |

14 | ||

23 |

T –Number = 23

T –Total = 4+5+6+14+23 = 52

The next part shows 4 more T –Shapes collected from a different position on the grid, these T –Shapes are from the second row:

10 | 11 | 12 |

20 | ||

29 |

T –Number = 29

T –Total = 82

11 | 12 | 13 |

21 | ||

30 |

T –Number = 30

T –Total = 87

12 | 13 | 14 |

22 | ||

31 |

T –Number = 31

T –Total = 92

13 | 14 | 15 |

23 | ||

32 |

T –Number = 32

T –Total = 97

Middle

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18

19

20

21

22

23

24

25

26

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29

30

31

32

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34

35

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37

38

39

40

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43

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47

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49

I’ve decided to take my first four T-Shapes from the 4th column and the 3rd row so I will start from the T –Number 18:

4 | 5 | 6 |

12 | ||

19 |

3 | 4 | 5 |

11 | ||

18 |

T –Number = 18 T –Number = 19

T –Total = 3+4+5+11+18 = 41 T –Total = 4+5+6+12+19 = 46

5 | 6 | 7 |

13 | ||

20 | ||

6 | 7 | 8 |

14 | ||

21 |

T –Number = 20 T –Number = 21

T –Total = 5+6+7+13+20 = 51 T –Total = 6+7+8+14+21 = 56

My next selection of four T –Shapes is from the 2nd column and 6th row so the starting T –Number would be 37:

22 | 23 | 24 |

30 | ||

37 | ||

23 | 24 | 25 |

31 | ||

38 |

T –Number = 37 T –Number = 38

T –Total = 22+23+24+30+37 = 136 T –Total = 23+24+25+31+38 = 141

24 | 25 | 26 |

32 | ||

39 | ||

25 | 26 | 27 |

33 | ||

40 |

T –Number = 39 T –Number = 40

T –Total = 24+25+26+32+39 = 146 T –Total = 25+26+27+33+40 = 151

The next step is to put the results into a table in order to see if there is any pattern amongst the results jus like in the 8*8 grid:

T -Number | T -Total |

18 | 41 |

19 | 46 |

20 | 51 |

21 | 56 |

T -Number | T -Total |

37 | 136 |

38 | 141 |

39 | 146 |

40 | 151 |

From both these tables I can see a pattern and that is that the T –Total increases by 5 every time the T –Number increases.

18, 19, 20, 21 37, 38, 39, 40

5 5 5 5 5 5

From this I am able to calculate a formula for this grid jus like for the 9*9 and 8*8 grid, to do this once again I will represent the numbers in one of the T-Shapes as “n”:

n-15 | n-14 | n-13 |

n-7 | ||

n | ||

23 | 24 | 25 |

31 | ||

38 |

Now to find out the formula I simply just add the T-Shape up:

T –Total = n+n-7+n-14+n-15+n-13 = 5n – 49 in order to check my equation I will once again pick a T –Number out of this 7*7 grid and then will substitute the T –Number into the equation as “n”:

25 | 26 | 27 |

33 | ||

40 |

T –Number = 40

T –Total = 25+26+27+33+40 = 151

The next step is to substitute 40 into 5n – 49 which equals T-Total = 5(40) – 49

= 200 – 49

= 151

I was correct and the equation for the 7*7 grid is 5n – 49. I will investigate on one more grid which is the 6*6 after doing so it will help me enable to find a relationship between the grids so doing a 6*6 grid will only help me get a more accurate answer to the relationship between the grid sizes.

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 |

I’ve decided to start from the T –Number 21 which is in the 3rd column and 4th row:

8 | 9 | 10 |

15 | ||

21 | ||

9 | 10 | 11 |

16 | ||

22 |

T –Number = 21 T –Number = 22

T –Total = 8+9+10+15+21 = 63 T –Total = 9+10+11+16+22 = 68

10 | 11 | 12 |

17 | ||

23 | ||

11 | 12 | 13 |

18 | ||

24 |

T –Number = 23 T –Number = 24

T –Total = 10+11+12+17+23 = 73 T –Total = 11+12+13+18+24 = 78

The next four T –Shapes will be collected from the T –Number 34 which is at the 2nd column and 6th row:

19 | 20 | 21 |

26 | ||

32 | ||

20 | 21 | 22 |

27 | ||

33 |

T –Number = 32 T –Number = 33

T –Total = 19+20+21+26+32 = 118 T –Total = 20+21+22+27+33 = 123

21 | 22 | 23 |

28 | ||

34 | ||

22 | 23 | 24 |

29 | ||

35 |

T –Number = 34 T –Number = 35

T –Total = 21+22+23+28+34 = 128 T –Total = 22+23+24+29+35 = 133

The next step is to put the results into tables:

T -Number | T -Total |

21 | 63 |

22 | 68 |

23 | 73 |

24 | 78 |

T -Number | T -Total |

32 | 118 |

33 | 123 |

34 | 128 |

35 | 133 |

From both these tables I can see a pattern and that is that the T –Total increases by 5 every time the T –Number increases.

21, 22, 23 ,24 32, 33, 34, 35

5 5 5 5 5 5

From this I am able to calculate a formula for this grid jus like for the 9*9, 8*8 and 7*7 grid, to do this once again I will represent the numbers in one of the T-Shapes as “n”:

n-13 | n-12 | n-11 |

n-6 | ||

n | ||

19 | 20 | 21 |

26 | ||

32 |

Now to find out the formula I simply just add the T-Shape up:

T –Total = n+n-6+n-12+n-11+n-12 = 5n – 42 in order to check if this equation is right I will jus use the T –Number from the T- Shape above and will substitute it into the equation 5n – 42.

19 | 20 | 21 |

26 | ||

32 |

T –Number = 32

T –Total = 19+20+21+26+32 = 118

The next step is substituting 32 into 5n – 42 equals = T –Total = 5(32) - 42

= 160 - 42

= 118

The equation is correct as I have managed to find the T –Total using the equation.

After collecting the equations for the four grid sizes the next step is to find and equation using the formulae of the grid sizes to achieve and equation which will enable us to find any T –Total from any grid size.

To do this I will begin by creating a table which will include all the equations from the four grid sizes:

Order of Grid | T -Number | Equations |

6 | N | 5N - 42 |

7 | N | 5N - 49 |

8 | N | 5N - 56 |

9 | N | 5N - 63 |

Looking at the table above, I can see that as the grid size increases by one the equation increases by 7 so therefore the grid size is in proportion to 7 which then helps to find an equation that will find any T –Total on any grid size:

So the equation is 5n – 7w, where w represents the grid size. In order to check if the equation is correct I will substitute the T –Shape below:

19 | 20 | 21 |

26 | ||

32 |

Conclusion

In a 9*9 grid the equation equals 5n -63 and the rotated equation equals 5n + 7 I can notice the difference between the equations which is 70. In an 8*8 the equation equals 5n -56 and the rotated equation is 5n + 7 the difference here is of 63 and in a 7*7 grid the equation is 5n – 49 and the difference is 56:

By using the difference between the equations and then comparing them with each other I am able to formulate an equation for the T –Total of a rotated T –Shape of a grid size

Grid Size: 7 8 9

56 63 70

- 7

From the above we can make the 7 a factor of the grid representing it as 7 as it is in proportion with the grid size because as the grid size increases by one the difference of seven also increases by seven so there fore leaving with an equation of 7(w +1) where w represents the grid size, this equation when added with the equation of a grid size helps to find the rotated T –Total of a T –Shape for example on a 9*9 grid I would like to find the rotated T –Total of the T –Number 23 to do this I simply substitute the T –Number into the equation below along with the grid size:

5n – 63 + 7 (w+1) therefore Rotated T –Total = 5(23) – 63 + 7(9+1)

= 115 – 63 + 70

= 52 + 70

= 122

16 | ||

23 | 24 | 25 |

34 |

4 | 5 | 6 |

14 | ||

23 |

T –Number = 23

T –Total = 4+5+6+14+23 = 52

T –Number = 23

T –Total = 23+24+25+16+34 = 122

As you can see the calculation is correct for finding a rotated 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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