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• Level: GCSE
• Subject: Maths
• Word count: 4177

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

Extracts from this document...

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

for the 9x9 grid.

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)

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

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

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