# t shape t toal

Extracts from this document...

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.

The 20 at the bottom of the T shape will be called the T-Number.

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.

T-19 | T-18 | T-17 |

T-9 | ||

T |

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

T-Total=5T-63

Time for me to check if this formula works:

T means T-Number.

T=20

T-Total=5x20-63

T-Total=100-63

T-Total=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-Total=112

T-Total=5x35-63

T-Total=175-63

T-Total=112

48 | 49 | 50 |

58 | ||

67 |

T-Total=272

T-Total=5x67-63

T-Total=335-63

T-Total=272

So now you know the formula for the 9 by 9 grid is T-Total=5T-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-Total=25

T-11 | T-10 | T-9 |

T-5 | ||

T |

Here is a converted version of the original. This is so I can work out a formula to find the T-total on a 5 by 5 grid.

Middle

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.

T | ||

T+5 | ||

T+9 | T+10 | T+11 |

T-Total=(T)+(T+5)+(T+10)+(T+9)+(T+11)

T-Total=5T+35

Now to check if the formula works:

3 | ||

8 | ||

12 | 13 | 14 |

T-Total=50

T-Total=5x3+35

T-Total=15+35

T-Total=50

Here is another check to make sure.

14 | ||

19 | ||

23 | 24 | 25 |

T-Total=105

T-Total=5x14+35

T-Total=70+35

T-Total=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.

T | ||

T+4 | ||

T+7 | T+8 | T+9 |

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

T-Total=(T)+(T+4)+(T+8)+(T+7)+(T+9)

T-Total=5T+28

Now to check if the formula works:

7 | ||

11 | ||

14 | 15 | 16 |

T-Total=63

T-Total=5x7+28

T-Total=35+28

T-Total=63

Here is another example for that formula:

14 | ||

18 | ||

21 | 22 | 23 |

T-Total=98

T-Total=5x14+28

T-Total=70+28

T-Total=98

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

T | ||

T+N | ||

T+2N-1 | T+2n | T+2N+1 |

N=Grid Width

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

T-Total=(T)+(T+n)+(T+2n)+(T+2n+1)+(T+2n+1)

T-Total=5T+7N

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 |

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.

T-6 | ||

T-2 | T-1 | T |

T+2 |

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

T-Total=5T-7

Let me try this formula on my T-Shape:

10 | ||

14 | 15 | 16 |

18 |

T-Total=73

T-Total=5x16-7

T-Total=80-7

T-Total=73

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

17 | ||

21 | 22 | 23 |

25 |

T-Total=108

T-Total=5x23-7

T-Total=115-7

T-Total=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 5T-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-Total of T-Shape=158

T-Total=5x33-7

T-Total=165-7

T-Total=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-Total of T-Shape=123

T-Total=5x26-7

T-Total=130-7

T-Total=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=5T-7n

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=5T+7N

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=5T+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=5T-7

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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