# Relationships with the T-shapes, the T-total and the position of the stair shape on the grid.

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Introduction

T- Total

By Gary Holmes

Introduction:

In this investigation in which I am presenting, I am going to look at the relationships with the T-shapes, the T-total and the position of the stair shape on the grid. I will be observing patterns and I will look at the other stairs to see if I can notice other formulas for these. Finally, I hope to be able to find the formula for any size grids and any 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 |

Look at the T-shape drawn on the 9 by 9 number grid.

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

This is called the T-total.

The number at the bottom of the T-shape is called theT-number.

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

T |

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.

Middle

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

X=5X33-42 (20+21+22+27+33=123

X=165-42

X=123

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+5T

X=7G+5T=7x3+5x2

=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=7x4+5x10 (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=7x9+5x2 (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=5x6-7

=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=5x11-7 (5+9+13+10+11=48)

=48

And 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=5x19-7 (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 |

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=5x4+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

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

Grey T-Total = 67 a difference of 42….. The formula 5T - 7G + 42

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-Total = 33

Grey T-Total = 82 a difference of 49….. The formula 5T - 7G + 49

7 by 7 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 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

T- Number = 16

Purple T-Total = 41

Grey T-Total = 97 a difference of 56….. The formula 5T - 7G + 56

Table of Results

Grid width | Purple T-Total | Grey 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 |

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 |

Now I will rotate it again 180° from it's original place

T- Number = 12

Purple T-Total = 25

Grey T-Total = 95 a 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

Grey 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

Grey T-Total = 129 a difference of 98….The formula 5T - 7G + 98

Conclusion: The amount of numbers in the T decide the number before N in my formula. The number before the W is found by

(T-total - 2T) divided by the top right number of the grid.

By Gary Holmes

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

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