# I have been set the task to find formulas that I need to find t the "T-Total" which is explained below:

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Introduction

Introduction

I have been set the task to find formulas that I need to find t the “T-Total” which is explained below:

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 |

Have a look at the T-shape highlighted on the nine by nine number grid above.

The total of the number in the T-shape is: 1+2+3+11+20=37

This is called the “T-Total” (which this time added up to 37).

The number at the bottom of the T-shape is called the “T-number”

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

For the task that I have been set I have to translate the T-shape to different parts of the grid and work out the T-Total. I have to find the relationships with the T-total and the T-number, and find out a formula.

To take this task further I will use grids of different sizes. Translate the shapes to different positions. Then I will work out the T-Total. Then I will find the relationship between the T-Total, T-number and the grid size, and find out a formula.

To take this task even further I will use grids of different sizes again. Try other transformations and combinations of different transformations. I will find the relationships between the T-Total, the T-numbers, the grid size and the transformations and find a formula.

Relationships between the T-Total and the T-number

9x9 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number.

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 |

The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number | 20 | 21 | 22 | 23 | 24 |

T-Total | 37 | 42 | 47 | 52 | 57 |

I will try and work out the formula by finding the difference of the numbers and producing a formula.

Middle

14

The T-total for this shape is 28 and the T-number is 14. I will use this to work out the formula for this grid.

28=5n-42

28=5x14-42

28=70-42

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 15 for the T-number where the total is 33.

33=5n-42

33=5x15-42

33=75-42

The formula also works for this.

6x6 Grid formula to find the T-Total = 5n-49

## The General Formula for different Grid sizes

I am going to find the relationship between the T-Total, the T-number and the grid size. I will try and find a formula to work out the T-Total from the T-number and the grid size.

### The Formulas that I have gathered

Grid Size (G) | Formula |

9x9 | T=5n-63 |

8x8 | T=5n-56 |

7x7 | T=5n-49 |

6x6 | T=5n-42 |

#### Each of these formulas are

T=5n-x

x= The number that is subtracted

In all of these formulas I have noticed that each of the subtractions made is the grid size multiplied by 7. I.e. For a 6x6 grid use the number 6. E.g. 5n-(7x6)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n-(7xG)

G= Grid size i.e. for a 6x6 grid use the number 6. E.g. 5n-(7x6)

Here are the results that I have gathered from the tests that I have done.

9x9

T-Number | 20 | 21 | 22 | 23 | 24 |

T-Total | 37 | 42 | 47 | 52 | 57 |

8x8

T-Number | 18 | 19 | 20 | 21 | 22 |

T-Total | 34 | 39 | 44 | 49 | 54 |

7x7

T-Number | 16 | 17 | 18 | 19 | 20 |

T-Total | 31 | 36 | 41 | 46 | 51 |

6x6

## T-Number | 14 | 15 | 16 | 17 |

T-Total | 28 | 33 | 38 | 43 |

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9 grid

5x20-(7x9)= 37

The formula has worked for the 9x9 grid

8x8 grid

5x18-(7x8)=34

The formula has worked for the 8x8 grid

7x7 grid

5x16-(7x7)=31

The formula has worked for the 7x7 grid

6x6 grid

5x14-(7x6)=28

The formula has worked for the 6x6 grid

The general grid size formula to find the T-Total = 5n-(Gx7)

Relationships between the T-Total and the T-number using rotations of 180°

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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 |

The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number | 2 | 3 | 4 | 5 |

T-Total | 73 | 78 | 83 | 88 |

I will try and work out the formula by finding the difference of the numbers and producing a formula.

Here is an example:

21 | 19 | 20 |

11 | ||

2 |

Here is the example shown as a formula:

n+ 19 | n+ 18 | n+ 17 |

n+9 | ||

n |

n= T-number

Below is my working out for the formula:

T= n+n+19+n+18+n+17+n+9

T= 5n+19+18+17+9

T= 5n+63

T= T-Total

n= T-number

The formula is T=5n + 63 for the 9x9 grid.

I will test this formula be using the T-shape below.

.

21 | 20 | 19 |

11 | ||

2 |

The t-total for this shape is 73 and the T-number is 2. I will use this to work out the formula for this grid.

73=5n+63

73=5x2+63

73=10+63

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 3 for the T-number where the total is 73.

78=5n+63

785x3+63

78=15+63

The formula also works for this.

9x9 Grid formula to find the T-Total = 5n+56

8x8 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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 |

The table below shows the results that I have gathered from the T-shapes on the 8x8 grid above.

T-Number | 2 | 3 | 4 | 5 |

T-Total | 66 | 71 | 76 | 81 |

I will try and work out the formula by finding the difference of the numbers and producing a formula.

Here is an example:

17 | 18 | 19 |

10 | ||

2 |

Here is the example shown as a formula:

n+ 15 | n+ 16 | n+ 17 |

n+8 | ||

n |

n= T-number

Below is my working out for the formula:

T= n+n+15+n+16+n+17+n+8

T= 5n+15+16+17+8

T= 5n+56

T= T-Total

n= T-number

The formula is T=5n + 56 for the 8x8 grid.

I will test this formula be using the T-shape below.

.

17 | 18 | 19 |

10 | ||

2 |

The t-total for this shape is 66 and the T-number is 2. I will use this to work out the formula for this grid.

66=5n+56

66=5x2+56

66=10+56

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 3 for the T-number where the total is 71.

71=5n+56

71=5x3+56

71=15+56

The formula also works for this.

8x8 Grid formula to find the T-Total = 5n+56

## The General Formula for different Grid sizes

I am going to find the relationship between the T-Total, the T-number and the grid size when the t-shape is rotated 180°. I will try and find a formula to work out the T-Total from the T-number and the grid size.

### The Formulas that I have gathered

Grid Size (G) | Formula |

9x9 | T=5n+63 |

8x8 | T=5n+56 |

#### Each of these formulas are

T=5n+x

x= The number that is added on

In all of these formulas I have noticed that each of the additions made is the grid size multiplied by 7. I.e. For an 8x8 grid use the number 8. E.g. 5n+(7x8)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n+(7xG)

G= Grid size i.e. for a 9x9 grid use the number 9. E.g. 5n+(7x9)

Here are the results that I have gathered from the tests that I have done.

9x9

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9

T-Number | 2 | 3 | 4 | 5 |

T-Total | 73 | 78 | 83 | 88 |

8x8

T-Number | 2 | 3 | 4 | 5 |

T-Total | 66 | 71 | 76 | 81 |

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9 grid

5x2+(7x9)= 66

The formula has worked for the 9x9 grid

8x8 grid

5x2+(7x8)=73

The formula has worked for the 8x8 grid

The general grid size formula to find the 180°

T-shape T-Total = 5n+(Gx7)

Relationships between the T-Total and the T-number using rotations of 90° Clockwise

9x9 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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 |

The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number | 10 | 11 | 12 | 13 |

T-Total | 57 | 62 | 67 | 72 |

I have worked out the formula as shown below:

n-6 | n+ 2 | n+ 10 |

n+1 | ||

n |

Conclusion

8x8 Grid

I am going to find the relationship between the T-Total and the T-number on an 8x8 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 45° clockwise.

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 |

The table below shows the results that I have gathered from the T-shapes on the 8x8 grid above.

T-Number | 25 | 26 | 27 | 28 |

T-Total | 76 | 81 | 86 | 91 |

I have worked out the formula as shown below:

n-23 | n- 14 | n- 5 |

n-7 | ||

n |

The formula is T=5n-49 for the 8x8 grid.

## The General Formula for different Grid sizes

I am going to find the relationship between the T-Total, the T-number and the grid size when the t-shape is rotated 45° clockwise. I will try and find a formula to work out the T-Total from the T-number and the grid size.

### The Formulas that I have gathered

Grid Size (G) | Formula |

9x9 | T=5n-56 |

8x8 | T=5n-49 |

#### Each of these formulas are

T=5n-x

x= The number that is added on

In all of these formulas I have noticed that each of the additions made is the grid size minus 1 multiplied by 7. I.e. For an 8x8 grid use the number 7. E.g. 5n-7(8-1)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n-7(G-1)

G= Grid size i.e. for a 9x9 grid use the number 9. E.g. 5n-7(9-1)

9x9 grid

5x28-7(9-1)= 84

The formula has worked for the 9x9 grid

8x8 grid

5x25-7(8-1)=49

The formula has worked for the 8x8 grid

The general grid size formula to find the clockwise 45°

T-shape T-Total = 5n-7(G-1)

##### A Table of Formulas that I have found

Translation | Formula |

None | 5n-(Gx7) |

180° | 5n+(Gx7) |

90° Clockwise | 5n+7 |

90° Anti-Clockwise | 5n-7 |

45°Clockwise | 5n-7(G-1) |

###### By Luke Scowcroft 10-0

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

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