# T-Total. I can work out a formula to find the T-total on a 9 by 9 grid.

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Introduction

Maths Coursework

T-Total

1.

Here is 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 |

The total of the numbers within the T shape is 1+2+3+11+20=37

I will call this the T-total. T would represent this.

The 20 at the bottom of the T shape will be called the T-Number. N would represent this.

1 | 2 | 3 |

11 | ||

20 |

Here is the T shape by itself. As you can see the middle column is going down in 9’s because the grid size is 9 by 9.

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

N-9 | ||

N |

Here is a converted version of the original. This is so I can work out a formula to find the T-total on a 9 by 9 grid.

T = N-19+N-18+N-17+N-9+N

T = 5N-63

Time for me to check if this formula works:

N means T-Number.

N = 20

T = 5x20-63

T = 100-63

T = 37

To make sure it is not a fluke, I will do 2 more checks on the same grid size.

16 | 17 | 18 |

26 | ||

35 |

T = 112

T = 5x35-63

T = 175-63

T = 112

48 | 49 | 50 |

58 | ||

67 |

T = 272

T = 5x67-63

T = 335-63

T = 272

So now you know the formula for the 9 by 9 grid is T =5N-63

2.

I will now find the formula for a 5 by 5 grid.

Here is the 5 by 5 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 |

The total of all the numbers inside the T-shape is 1+2+3+7+12=25

As you can see, the middle column is going down in 5’s because of the grid size.

1 | 2 | 3 |

7 | ||

12 |

T = 25

N-11 | N-10 | N-9 |

N-5 | ||

N |

Here is a converted version of the original.

Middle

The total amount of numbers inside the T-Shape is 3+8+12+13+14=50

The 3 at the top of the T-Shape will be called the T-Number.

3 | ||

8 | ||

12 | 13 | 14 |

Here is the T-Shape by itself. As you can see the numbers in the middle is going up in 5’s because the grid width is 5.

Here is the converted version so I can work out the formula for the T-Total.

N | ||

N+5 | ||

N+9 | N+10 | N+11 |

T = (N)+(N+5)+(N+10)+(N+9)+(N+11)

T = 5N+35

Now to check if the formula works:

3 | ||

8 | ||

12 | 13 | 14 |

T = 50

T = 5x3+35

T = 15+35

T = 50

Here is another check to make sure.

14 | ||

19 | ||

23 | 24 | 25 |

T = 105

T = 5x14+35

T = 70+35

T = 105

Now I will see another formula for another grid size, which is 4 by 6.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

That will be the T-Shape I will be finding the T-Total formula.

Firstly here is the formula I need to use.

N | ||

N+4 | ||

N+7 | N+8 | N+9 |

That is the converted version of the 4 by 6 grid size.

T = (N)+(N+4)+(N+8)+(N+7)+(N+9)

T = 5N+28

Now to check if the formula works:

7 | ||

11 | ||

14 | 15 | 16 |

T = 63

T = 5x7+28

T = 35+28

T = 63

Here is another example for that formula:

14 | ||

18 | ||

21 | 22 | 23 |

T = 98

T = 5x14+28

T = 70+28

T = 98

Now I will figure out a formula to find the T-Total for an inverted T-Shape on any size grid.

N | ||

N+G | ||

N+2G-1 | N+2G | N+2G+1 |

G = Grid Width

Here is the full formula to work when solving out the formula for an inverted T-Shape on any size grid.

T = (N)+(N+G)+(N+2G)+(N+2G+1)+(N+2G+1)

T = 5N+7G

Now lets try that formula on the 5 by 5 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 |

T = 5N+7G

T = 5N+7x5

T = 5N+35

As you can see if you go up to page 5-6 you will find that it works out as the same formula as the one on this page.

Now for a different sized grid I haven’t used before.

I will use a 3 by 7 grid size.

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

10 | 11 | 12 |

13 | 14 | 15 |

16 | 17 | 18 |

19 | 20 | 21 |

T = 5N+7G

7G = 7xGrid width

T = 5N+7x3

T = 5N+21

To make sure this formula works I will find an inverted T-Shape on the 3 by 7 grid and see if that formula works.

8 | ||

11 | ||

13 | 14 | 15 |

T = 61

T = 5x8+21

T = 40+21

T = 61

Now that has been completed I will convert the T-Shape 90° to the right.

Here is a 5 by 5 grid to test on.

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 |

The Total amount of numbers inside the T shape is 11+12+13+8+18=62

The 11 at the left of the T-Shape will be called the T-Number.

8 | ||

11 | 12 | 13 |

18 |

Here is the T-Shape by itself. As you can see the numbers on the right are going down in 5’s due to the grid size, which is 5.

Here is the converted version so I can work out the formula for the T-Total.

N-3 | ||

N | N+1 | N+2 |

N+7 |

T = (N)+(N+1)+(N+2)+(N-3)+(N+7)

T = 5N+7

Now to check if this formula works:

8 | ||

11 | 12 | 13 |

18 |

T = 62

T = 5x11+7

T = 55+7

T = 62

Here is another T-Shape to make sure:

15 | ||

18 | 19 | 20 |

25 |

T = 97

T = 5x18+7

T = 90+7

T = 97

Here is another grid size to find out what formula I need to use for that specific grid size.

The grid size I’ll use will be 7 by 4.

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 |

Firstly here is the formula I’ll need.

N-5 | ||

N | N+1 | N+2 |

N+9 |

T = (N)+(N+1)+(N+2)+(N-5)+(N+9)

T = 5N+7

Now to check if the formula works:

5 | ||

10 | 11 | 12 |

19 |

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 |

This will be the T-Shape I will be using to find out if my formula works. I will firstly need to figure out a formula.

N-6 | ||

N-2 | N-1 | N |

N+2 |

T = (N)+(N-1)+(N-2)+(N-6)+(N+2)

T = 5N-7

Let me try this formula on my T-Shape:

10 | ||

14 | 15 | 16 |

18 |

T = 73

T = 5x16-7

T = 80-7

T = 73

Now let’s try this on another T-Shape in the same grid size:

17 | ||

21 | 22 | 23 |

25 |

T = 108

T = 5x23-7

T = 115-7

T = 108

Now this has been completed I will now try to find a formula to all T-Shapes rotated 270° to the right on any size grid. Although I should solve the full working out, I found that the formula 5N-7 could be worked for all grid sizes. Here is an example on a 9 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 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

T = 158

T = 5x33-7

T = 165-7

T = 158

Here is another example:

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 |

T = 123

T = 5x26-7

T = 130-7

T = 123

So now you understand that for the first T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5N-7G

Now you understand that for the inverted T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5N+7G

Now you understand that for the first sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5N+7

Now you understand that for the second sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5N-7

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

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