# T-Totals.Plan: Part 1 I will investigate the relationship between the T-total and the T-number on a 9x9 grid. Part 2 I will investigate the relationship between the T-total, T-number and grid size.

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Introduction

T - Totals Investigation

T-Totals

Introduction:

This is an investigation on T-totals. By starting with a 9x9 grid, and numbering it, so the top left corner numbered with one and work downwards. An example of the grid is shown below.

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 investigation will be done with a standard T-shape which is 3 along and from the centre 2 more down. As shown below.

2 | 3 | 4 |

12 | ||

21 |

This is the t-number for a T-shape.

The T-total is all the numbers in the T-shape added together. The T-total for this specific example is: 2+3+4+12+21=42.

Plan:

Part 1 – I will investigate the relationship between the T-total and the T-number on a 9x9 grid.

Part 2 – I will investigate the relationship between the T-total, T-number and grid size.

Part 1:

Diagrams:

20 | 21 | 22 |

30 | ||

39 | ||

52 | 53 | 54 |

62 | ||

71 |

20+21+22+30+39=13252+53+54+62+71=292

T-total = 132 T-total = 292

4 | 5 | 6 |

13 | 14 | 15 |

22 | 23 | 24 |

4+5+6+14+23=52

T-total = 52

From now on:

Let T-total be T!

Let T-number be n!

Now I have shown some examples, a table is required so the formula can be solved between the relationship of the T and n for a 9x9 grid.

n T

1 X

X

19 X

20 37

21 42

22 47

23 52

24 57

25 62

26 67

27 X

28 X

29 82

38 127

80 337

Middle

5n 100 105 110 115 120 125

T 37 42 47 52 57 62

5n to T - 63 - 63 - 63 - 63 - 63 - 63

Therefore the table above shows the formula will have – 63 in, because – 63 is a constant difference.

The formula comes out to be 5n – 63 = T for a 9x9 grid only.

As a test if n = 38

With the formula: (5 x 38) – 63 = T

- – 63 = T

127 = T

This is correct; the table shows that the formula works.

I predict that if the T-number is 40, with the formula of 5n – 63 = T for a 9x9 grid, T will equal: 137

T = 5n – 63

= (5 x 40) – 63

= 200 – 63

T = 137

To prove that the T-total for the T-number of 40 is 137:

21 | 22 | 23 |

31 | ||

40 |

40 + 31 + 22 + 23 + 21= T

137 = T

So the formula is correct as the predicted T-total with the formula was correct, but only for a 9x9 grid.

I think that the formula will also work with incomplete T-shapes too, if 19 is used as n, I can use the formula obtained to prove it:

T = (5 x 19) – 63

= 95 – 63

T = 32

To show that this is correct:

1 | 2 |

10 | |

19 |

T=19+10+1+2

T= 32

It is correct the formula also got 32, showing the formula works for T-shapes even if they are not complete.

Part 2:

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 |

Conclusion

I think that I have found the relationship between grid size, T-number and T-total. Because of the table at the top of the page, shows how found that there is always 5n in the formula and then the grid size is multiplied by -7 to give T. So I think the formula is:

5n – 7G = T

To test the new formula of T = 5n -7G, I will have the grid size of 10 and n being 24.

The formula gets: T = (5 x 24) – (7 x 10)

T = 120 – 70

T = 50

To check this:

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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

T= 3+4+5+14+24= 50

Excellent, the formula obtained which links the T-total, T-number and grid size is:

T = 5n – 7G

Whilst doing this investigation a number of questions sprung to mind, what would happen if the T-shape was turned round, through 90o, 180o and 270o? I also wondered, what would happen to the T-total once the T-shape had been rotated around a number in the grid, how this would relate to the new T-total, the old T-number and the number the T-shape is rotated around? How would the T-total change when the T-shape is reflected? How would the T-total change when it is enlarged or elongated? I also thought that what would happen to the T-total if the T-shape is translated about the grid?

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

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