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T-Total Coursework

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Introduction

Kamran Adnan (10 S2)        Maths Coursework        T-Total

Part 1

Here we have a grid with nine numbers in a row starting from 1 and ending with 81. There you can see the shape, which is marked in red colour. This shape is called a T-Shape. The T-Shape is always a shape of five numbers.

 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 number at the bottom is 20. This number is called the T-Number. The sum of all the five numbers is called the T-Total. In this case the T-Number would be:

1 + 2 + 3 + 11 + 20 = 37

To find out a relationship between the T-Number and the T-Total we need more results in the same 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

In this grid we have six more T-Shapes. Now we need to find out their T-Totals.

1 + 2 + 3 + 11 + 20 =        37

2 + 3 + 4 + 12 + 21 =         42

3 + 4 + 5 + 13 + 22 =        47

4 + 5 + 6 + 14 + 23 =        52

Looking at the results we can see that if the T-Number goes one up the T-Total goes five up. This is because by raising the T-Number by one, each number of the T-Shape will go one up. Because there are five numbers in a T-Shape the T-Total goes 5*1 = 5 steps up.

The formula for finding out the T-Total by using the T-Number is 5*N – 63. The following is an explanation how I found this formula out.

I already knew the T-Total for the T-Number 20. As we can see in the results is that every time the T-Number goes up one, the T-Total goes up five. So I had to multiply 20 by 5 because the T-Total goes five up if the T-Number goes one up.

Middle

49

50

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52

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67

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99

1 + 2 + 3 + 13 + 24 = 43

2 + 3 + 4 + 14 + 25 = 48

3 + 4 + 5 + 15 + 26 = 53

4 + 5 + 6 + 16 + 27 = 58

We can see again that if the T-Number goes one up the T-Total goes five up. This is because by raising the T-Number by one, each number of the T-Shape will go one up. Because there are five numbers in a T-Shape the T-Total goes 5*1 = 5 steps up.

This means again that the formula will have to start with 5*T-Number and end with a subtraction of a number that we will find now.

To find this subtrahend we have to methods. I will start with adding the differences between the T-Number and the four other numbers in the T-Shape (T-Number is 24):

(24 – 13) + (24 – 3) + (24 – 2) + (24 – 1)

= 11 + 21 + 22 + 23

= 77

The second method would be as followed (T-Number is 24):

5*24 – 43 = 77

I got the number 43 by adding the numbers of the T-Shape with the T-Number 24. For this I just looked up to the results.

Now I am going to test this formula.

 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

The T-Number 98

98*5-77 = 413

Check: 75 + 76 + 77 + 87 + 98 = 413

The T-Number 32

32*5-77 = 83

Check: 9 + 10 + 11 + 21 + 32 = 38

The T-Number 70

70*5 – 77 = 273

Check: 47 + 48 + 49 + 59 + 70 = 273

The T-Number 61

61*5 - 77 = 228

Check: 38 + 39 + 40 + 50 + 61 = 228

Next I am going to use one more grid, which will be smaller than the first one. I will use a grid with only 4 numbers in a row.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

I will start again with the T-Shape that has the smallest T-Number. In this grid it will be the T-Number 10. The T-Total for this T-Shape will be:

1 + 2 + 3 + 6 + 10 = 22

The formula for this shape will again start with 5*T-Number because the T-Shape still has five numbers and if each number goes one up the T-Total will have to go five up. The last number I will again find out by calculating the followings:

5*10 = 50

50 – 22 = 28

Therefore the formula has to be 5*T-Number – 28.

To check whether this formula is correct I will need some more results.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

T-Number 11

11*5-28 = 27

Check: 2 + 3 + 4 + 7 + 11 = 27

T-Number 18

18*5-28 = 62

Check: 9 + 10 + 11 + 14 + 18 = 62

After finding out the formula for each of the grids I looked at the formulae. These were:

9        5* N - 63

11        5* N -77

4        5* N -28

By looking at these formulae I saw that always the last number is a factor of 7. By dividing each number by 7 you get the amount of numbers in each line.

63/7 = 9

77/7 = 11

28/7 = 4

Therefore the general formula for finding the T-Total by using the T-Number is:

T = 5*N – (G*7)

E.G. the formula for the T-Total for a grid with 9 columns will be:

T = 5*N – (9*7)

T = 5*N – 63

There you can see that this is the same formula as I wrote and proved for the first grid, which had 9 columns.

PART 3

180° rotated T-Shape

In this section there is change in the size of grid. There is also transformation and combinations of transformations. I need to find out the investigation of the relationship between the T-Total, the T-Number, the grid size and the transformations.

If I turned the T-Shape around 180 degrees it would look like this.

 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

Once I have done this I realized that if I reverse the T-Shape I should have to reverse something in the formula.

The formula has a multiplication and a subtraction sign in it and that one of these two has to be turned into the negative. It cannot be the multiplication because in this case the T-Total would be smaller than the T-Number, whereas it is not possible to have a smaller T-Total than the T-Number. Therefore I think I have to turn the subtraction sign into an addition sign. After this the formula should be:

5*N – 63  →  5*N + 63

Now I need to check this formula.

The T-Number will be the number at the top of the T-Shape.

The formula would be:

5*2+63 = 73

Check: 2 + 11 + 19 + 20 + 21 = 73

The formula has worked.

A general formula for finding the T-Total in a 90° clockwise rotated shape this would be:

5*N + 7G

This is because the last number occurs by multiplying the grid size by seven.

270° rotated T-Shape

Next I will find out the formula for a 270° clockwise rotated T-Shape.

 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

We need to add up all the differences between the T-Number and all the other numbers in the T-Shape. This would be:

12 – 11 = 1

12 – 10 = 2

12 – 1 = 11

12 – 19 = -7

1 + 2 + 11 – 7 = 7

The result of adding up all these differences is 7. This means the formula for a T-Shape, which is 270° clockwise rotated, is 5*N – 7

To prove this I am going to check it on the following shape:

 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

Conclusion

To check that this formula is correct I am going to show this in a grid. Here the T-Shape has been rotated on the point 67:

 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

Now I am going to use the formula and put in the original numbers.

(N + a – bG) + (b - aG)

=        (38 + 2 + 3*9) + (3 – 2*9)

=        40 + 27 + 3 – 18

=        67 – 15

=        52

With this formula we get to the new T-Number in a 90° rotation. Next we need to find and add the formula for a 90° clockwise rotated shape.

We can find this formula easily by doing followings:

 N + 2 – 2G N N + 1 N + 2 N + 2 + 2G

We need to add up each formula in this T-Shape above.

N

N + 1

N + 2

N + 2 – 2G

+        N + 2 + 2G

_        .

=        5N + 7

So the final formula for a 90° clockwise rotation is

5(N + a – bG) + (b - aG) + 7

Conclusion

In this project we have found out many ways in which we can solve a problem that we might have with the T-Shape. It can be in various different positions with different grid sizes. We made the calculation easier by creating a main formula, which can change for all the different circumstances.

The followings are all the formulae I found out and proved in this project. They can work for any grid size.

Normal T-Shape                5N - 7G

90° clockwise rotated                5N + 7

180° clockwise rotated        5N + 7G

270° clockwise rotated        5N - 7G

Translation of T-Shape        (N + x – yG)*5 – 7G

90° rotation                        5(N + a – bG) + (b - aG) + 7

Page

T = T-Total; N = T-Number; G = Grid size;

x = horizontal movement; y = vertical movement (for translation)

a = horizontal movement; y = vertical movement (for rotation)

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

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