# This is an investigation to find a relationship between the T-totals and the T-number. The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the

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Introduction

Lianne Haley

COURSEWORK INVESTIGATION

T-Totals

The Problem, the Plan and possible extensions

This is an investigation to find a relationship between the T-totals and the T-number.

The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the top row 1-9.

The diagram shown has an upright T-shape, the total of the numbers inside the T-shape is 1+2+3+11+20 = 37, and this is called the T-total. The number at the bottom of the T-shape is called the T-number. The T-number, for the example T-shape given, is 20.

I need to be systematic in my approach so initially I will be investigating the relationship between the T-totals and T-numbers when the T-shape translates on the 9x9 grid, starting with the 1st row then the 2nd row etc. This will keep it simple for me to spot any patterns.

Later I will be investigating T-shapes on different sized grids, again translating the T-shape to different positions on the grids to find a relationship between the T-totals and the T-numbers.

I can also use grids of different sizes again and try other transformations and combinations of transformations and investigate relationships between the T-totals, the T-numbers, the grid size and the transformations.

Hypothesis

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 |

Middle

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

KEY | T-Shape Numbers | T-Numbers | T-Totals | Differences Between T-Totals |

1+2+3+8 | 14 | 28 | 5 | |

2+3+4+9 | 15 | 33 | 5 | |

3+4+5+10 | 16 | 38 | 5 | |

4+5+6+11 | 17 | Predict: 43 Calculation shows to be correct | Predict: 5 |

Again the T-Totals increase by +5.

I will check again.

6x6 Grid Row3 (13-18)

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 |

KEY | T-Shape Numbers | T-Numbers | T-Totals | Differences Between T-Totals |

13+14+15+20 | 26 | 88 | 5 | |

14+15+16+21 | 27 | 93 | 5 | |

15+16+17+22 | 28 | 98 | 5 | |

16+17+18+23 | 29 | Predict: 103 Calculation shows to be correct | Predict: 5 |

I have jumped to Row 3 to prove my hypothesis was correct and it was.

So if my calculations of +5 are the same for a 9x9 grid and 6x6 grid it will be the same for an 8x8 grid and a 14x14 grid.

I will tabulate my results.

8x8 Grid Row 4 (25-32)

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 |

KEY | T-Shape Numbers | T-Numbers | T-Totals | Differences Between T-Totals |

25+26+27+34 | 42 | 154 | 5 | |

26+27+28+35 | 43 | 159 | 5 | |

27+28+29+36 | 44 | 164 | 5 | |

28+29+30+37 | 45 | 169 | 5 | |

29+30+31+38 | 46 | 174 | 5 | |

30+31+32+39 | 47 | Predict: 179 Calculation shows to be correct | Predict: 5 |

14x14 Grid; Row 9 (113-126)

99 | 100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 |

113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 |

127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 |

141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 |

155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 |

KEY | T-Shape Numbers | T-Numbers | T-Totals | Differences Between T-Totals |

113+114+115+128 | 142 | 612 | 5 | |

114+115+116+129 | 143 | 617 | 5 | |

115+116+117+130 | 144 | 622 | 5 | |

116+117+118+131 | 145 | 627 | 5 | |

117+118+119+132 | 146 | 632 | 5 | |

118+119+120+133 | 147 | 637 | 5 | |

119+120+121+134 | 148 | 642 | 5 | |

120+121+122+135 | 149 | 647 | 5 | |

121+122+123+136 | 150 | 652 | 5 | |

122+123+124+137 | 151 | 657 | 5 | |

123+124+125+138 | 152 | 662 | 5 | |

124+125+126+139 | 153 | Predict: 667 Calculation shows to be correct | Predict: 5 |

My results confirm that with any number grid the T-Totals increase by +5.

Testing the Formula

With my results recorded on the table above for the 14x14 grid I will check my original formula to make sure this is correct.

n = 144

t = 5n-63

t = (5x144)-63

t = 720-63

t = 657 Incorrect

n = 152

t = 5n-63

t = (5x152)-63

t = 760-63

t = 697 Incorrect

This shows me that different sized number grids require a different formula as predicted in my hypothesis.

In this case:

n = 152

t = 662 (123+124+125+138) +n

123 | 124 | 125 |

138 | ||

152 |

152-123=29, 152-124=28, 152-125=27, 152-138=14

So to work out the T-Total, this simplifies to:

t = ((n-29) + (n-28) + (n-27) + (n-14) +n)

123 | 124 | 125 |

138 | ||

152 |

n-29 | n-28 | n-27 |

n-14 | ||

n |

this simplifies to:

t = 5n-98 (98 being the total of 29+28+27+14)

The formula is dependent on the size of the grid.

On a 9x9 grid the formula is t = 5n-63 63 is equal to 7x9

On a 14x14 grid the formula is t = 5n-98 98 is equal to 7x14

So on a 6x6 grid would the formula be: t = ((5n-(7x6))

Results

I will work this out using the table previously shown:

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 |

KEY | T-Shape Numbers | T-Numbers | T-Totals | Differences Between T-Totals |

13+14+15+20 | 26 | 88 | 5 | |

14+15+16+21 | 27 | 93 | 5 | |

15+16+17+22 | 28 | 98 | 5 | |

16+17+18+23 | 29 | 103 | 5 |

n = 29

t = ((5n-(7x6))

t = 5n-42

t = (5x29)-42

t = 145-42

t = 103 Correct

I will let g = number grid size

E.g. an 8x8 grid would be:

g = 8

So the formula will now be:

t = 5n-7g

I have an 8x8 grid, what is the T-Total if the T-Number is 46?

n = 46

t = 5n-7g

t = (5x46)-(7x8)

t = 230-56

t = 174 Correct

See below an excerpt taken from my results from the 8x8 grid, see page 7.

29+30+31+38 | 46 | 174 | 5 | 128 | 4 |

Conclusion

With a 90°clockwise transformation of an upright T the T-Number is neither the lowest nor the highest, but the highest number minus the T-Number is equal to 7.

31 | 32 | 33 |

40 | 41 | 42 |

49 | 50 | 51 |

With a 270° clockwise transformation, the T-Number is, again, neither the highest nor the lowest number but the T-Number minus the lowest number is equal to 7.

31 | 32 | 33 |

40 | 41 | 42 |

49 | 50 | 51 |

Justifying the Formula

Question

I need to find out the T-Total for a 270° T-Shape who’s T-Number is 55 on an 11x11 grid.

Answer

n=45

t=5n+7

t= (5x45) +7

t=225+7

t=232

Check

36 | ||

45 | 46 | 47 |

58 |

Conclusion

I have found that each different translation requires a formula similar to each other translation for different sized grids.

I believe I have fully justified the explanation of this occurrence by checking my findings and tabulating results.

My initial hypothesis was correct stating that each T-Shape would increase by 5 thus finding the relationship between the T-Number and T-Total was +5 for any translation on any sized grid.

Further Extensions

If I had time I could explore the relationship between different sized T-Shapes on different sized grids, for example, extended T-Shapes (5 on the top row and 3 on the bottom row etc.) and elongated T-shapes.

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

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