# T-totals, T-numbers and T-shapes.

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Introduction

Introduction

Recently, we have been set a task to investigate about T-totals, T-numbers and T-shapes. T-shapes consist of 5 numbers; these 5 numbers will form a T-total when added together. A T-number is the bottom number of a T-shape.

This piece of coursework will consist of 3 parts. We will have to try and investigate what the relationship is between the T-total, T-number and the T-shape in the time provided to complete this piece of coursework. Various methods will have to be used to find the formula; these various methods will consist of things such as algebra, trial and improvement, generalization etc. Different grid sizes would be used in order to find the “Master Equation” of T-totals, T-numbers and T-shapes.

An example of a T-shape and what it consists of.

x | x | x |

x | ||

x |

This is a T-shape.

When all the numbers are added up together it would form a T-total.

X+X+X+X+X=T-total.

Along this row is the T-number.

Part 1

Using T-shapes, T-totals and T-numbers I am going to investigate the relationship between the T-total and the T-number. To do this I need a formula relating the T-total and T-number.

I am going to use a 9x9 grid

Middle

Firstly I am going to use an 8x8 grid, reason being that I am going to start off with small grids then work up to larger grids.

8x8 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 |

Below I have chosen my examples of T-shapes that I am going to experiment with.

1 | 2 | 3 |

10 | ||

18 |

The T-total of the numbers in this T-shape is:

1+2+3+10+18=34.

The T-number is 18.

2 | 3 | 4 |

11 | ||

19 |

The T-total of the numbers in this T-shape is:

2+3+4+11+19=39.

The T-number is 19.

3 | 4 | 5 |

12 | ||

20 |

The T-total of the numbers in this T-shape is:

3+4+5+12+20=44.

The T-number is 20.

4 | 5 | 6 |

13 | ||

21 |

The T-total of the numbers in this T-shape is:

4+5+6++13+21=49.

The T-number is 21.

5 | 6 | 7 |

14 | ||

22 |

The T-total of the numbers in this T-shape is:

5+6+7+14+22=54.

The T-number is 22.

6 | 7 | 8 |

15 | ||

23 |

The T-total of the numbers in this T-shape is:

6+&+8+15+23=59.

The T-number is 23.

Using my examples I have worked out that a pattern is starting to form.

T-number | T-total |

18 | 34 |

19 | 39 |

20 | 44 |

21 | 49 |

22 | 54 |

23 | 59 |

52 | 204 |

Prediction

Every time a T-number goes up by 1, The T-total goes up by 5.

35 | 36 | 37 |

44 | ||

52 |

The T-total of the numbers in this T-shape is:

35+36+37+44+52=204.

The T-number is 52. Correct

The equation for an 8x8 grid is 5N-56.

N-17 | N-16 | N-15 |

N-8 | ||

N |

N+ (N-8)

Conclusion

I have proved this above.

Expressing result algebraically:

N-19 | N-20 | N-21 |

N-10 | ||

N |

T-total= N+ (N-10) + (N-20) + (N-21) + (N-19)=5N-70

Proof that my equation does work.

1 | 2 | 3 |

12 | ||

22 |

T-total=40

40=5x-70

(5x22)=110

40=110-70

110-70=40

Master equation for Part 2:

5x-(grid size x7) =T-total

Proof of Master equation for grid size 8

46 | 47 | 48 |

55 | ||

63 |

5x63-(8x7) =259

46+47+48+55+63=259 Correct.

Proof of Master equation for grid size 8

71 | 72 | 73 |

82 | ||

92 |

5x92-(10x7) =390

71+72+73+82+92=390 Correct.

Part 3

Now instead of just shifting the directions of the T-shape I am going to rotate them, by this I mean turning the T-shape upside down and moving it in opposite directions. But I am still going to have to investigate the relationships between the T-total, the T-numbers, the grid size and the Transformations.

I am going o start off with grid size 11.

11x11 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 |

68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 |

78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 |

89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |

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

111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 |

Below I have chosen my examples of T-shapes that I am going to experiment with.

First of all I am going to turn my T-shapes facing left.

70 | ||

78 | 79 | 80 |

91 |

The T-total is:

70+78+79+80+91=398

The T-number is 78.

I was unable to finish off Part 3 due to running out of time. But if I were able to complete Part 3 I would have got examples of many T-shapes facing different directions up, down, left and right. I would have also put in many results tables and included lots of information of all sort of things, from examples to expressing results algebraically.

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

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