# T-Shapes.I am going to investigate the T-Shapes grid. I will be first looking at the 9 x 9 grid. I will then show how many T-shapes I would be able to make, by only moving the shapes across only once each time.

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Introduction

T-Shapes

I am going to investigate the T-Shapes grid. I will be first looking at the 9 x 9 grid.

I will then show how many T-shapes I would be able to make, by only moving the shapes across only once each time.

Once I have done this I will add up all the T-Shapes to find out the T-Total.

The last number at the bottom of the T-Shape is called the T-number.

The T-total for this T-shape is:

1 + 2 + 3 + 11 + 20 = 37

37 = T-Total

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 T-number for this T-shape is 20.

1 | 2 | 3 |

11 | ||

T = 20 |

If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.

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

T-9 | ||

T |

## You will notice that the centre column of the T-Shape is going up in 9’s because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:

## (T-19) + (T-18) + (T-17) + (T-9) + T = T- total

## = 5T - 63

Now to test this formula to see if it works

For T-Total I will use the letter X

Middle

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## X = 5 x 33 - 42 (20 + 21 + 22 + 27 + 33) = 123

X = 165 - 42

X = 123

Part 2

Now I have worked the formula out for the T-Shape in the one position, I am going to translate the T-Shape to different positions and investigate the relationships between the T-Total, the T-number and the grid size.

A 3 by 3 grid can be used for the other 3.

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

T | ||

T+G | ||

T+2G+1 | T+2G | T+2G-1 |

## T

T + G

T + 2G

T + 2G -1

## T + 2G + 1

7G + 5 T

## X = 7G + 5T = 7 x 3 + 5 x 2

## = 21 + 10 (2 + 5 + 7 + 8 + 9 = 31)

= 31

I will now try this new formula on a 4 by 7 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 |

7G + 5T

X = 7 x 4 + 5 x 10 (17 + 18 + 19 + 14 + 10) = 78

= 28 + 50

= 78

I will now try it on a 8 by 3 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 |

7G + 5T

X = 7 x 9 + 5 x 2 (19 + 20 + 21 + 11 + 2) = 73

= 56 + 10

= 73

I have tried this new formula for an inverted T and it works, the formula is: 7G + 5T

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

T-1-G | ||

T-2 | T-1 | T |

T-2+G |

## T

T-1

T-2

T-2-G

## T-2 + G

## 5T - 7

## X = 5T -7 = 5 x 6 - 7

## = 30 – 7 (1 + 4 + 7 + 5 + 6) = 23

= 23

I will now try the new formula on a 4 by 4 grid

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

5T - 7

X = 5 x 11 -7 (5 + 9 + 13 + 10 + 11) = 48

= 48

On a 4 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 |

5T - 7

X = 5 x 19 -7 (13 + 17 + 21 + 18 + 19) = 88

= 88

I have tested this new formula for a side ways T the formula works it is: 5T-7

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

This is the minimum t-shape possible that you can get on any grid size.

T+2-G | ||

T | T+1 | T+2 |

T+2+G |

## T

T + 1

T + 2

T + 2 + G

## T + 2 - G

5t + 7

The formula is: 5T + 7

## X = 5T + 7 = 5 x 4 + 7

## = 20 +7 (3 + 6 + 9 + 5 + 4) = 27

## = 27

I will now try the new formula on a 6 by 6 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 |

Conclusion

T- Number = 14

Purple T-Total = 33

Blue T-Total = 82 a difference of 49

The formula 5T – 7G + 49

I will not try out a 7 by 7 width Grid to see if my formula works

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 |

T- Number = 16

Purple T-Total = 41

Blue T-Total = 97 a difference of 56

The formula 5T – 7G + 56

### Table of Results

Grid width | Purple T-Total | Blue T-Total | Difference between the two T-Total |

5 | 25 | 67 | 42 |

6 | 33 | 82 | 49 Goes up in 7’s |

7 | 41 | 97 | 56 |

Now I will rotate it again 180° from its original place

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- Number = 12

Purple T-Total = 25

Blue T-Total = 95 a of difference of 70

The formula 5T – 7G + 70

Now I’m going to see what happens on a 6 by 6 width 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 |

T- Number = 14

Purple T = 28

Blue T = 112

A difference of 84 The formula 5T – 7G + 84

7 by 7 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 |

T- Number = 16

Purple T-Total = 31

Blue T-Total = 129 a difference of 98

The formula 5T – 7G + 98

### Table of Results

Grid width | Purple T-Total | Blue T-Total | Difference between the two T-Total |

5 | 25 | 95 | 70 |

6 | 28 | 112 | 84 Goes up by 14 |

7 | 31 | 129 | 98 |

Conclusion

From my analysis I have found that as long as the translation is the same the formula works, no matter if you enlarge the grid size or make it smaller. Also the number has to be a antigen number which you can multiply so that the formula works, if you rotate the T shape the formula will not work.

Sara Ali-Asghar GCSE Maths – Page of 2003

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

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