# T-total Investigation

Extracts from this document...

Introduction

Habibur Rahman Maths Coursework

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 |

On the grid on the right, you can see a 9 by 9 grid. On the grid, we see a “T” shape highlighted. The sum of the numbers within the T-shape is 1+2+3+11+20 = 37. This is known as the T-total.

The T-number is the number that is at the bottom of the T-shape. In this example, 20 is the T-number.

During this coursework, I will be investigating the relationships between the T-shapes and how they relate to grid size. I will also be looking closely into the significance of the T-number and how it could be used to figure out the T-total.

9 by 9 Grid

We have already figured out the t-total for one t-shape in the 9 by 9 grid. Here are some more results.

34+35+36+44+53 = 202

46+47+48+56+65 = 262

5+6+7+15+24 = 57

58+59+60+68+77 = 322

In this investigation, I’ll be implementing the use of equations. Here is how I started off.

1 | 2 | 3 |

10 | 11 | 12 |

19 | 20 | 21 |

If I bring all these figures together, I should get a correct equation.

T = N -19+N -18+N -17+N -9+N

= 5N – 63

Middle

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

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46

47

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49

50

51

52

53

54

55

56

57

58

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61

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63

64

As I did with the 9 by 9 grid, I worked out the t-total for various t-shapes in the grid the long way (adding each individual digit in the t-shape, one by one). I will the see

if the equation for the 9 by 9 grid would work on this

grid.

1+2+3+10+18 = 34

5+6+7+14+22 = 54

33+34+35+42+50 = 194

37+38+39+46+54 = 214

Then I attempted to use the equation in the previous grid to work out the t-total of a t-shape.

T = 5N - 63

N = 22

T = (5x22) – 63

= 110 –63

= 47

As we can see above, the result was wrong. So I decided to do what I did in the previous grid and work out a new equation.

1 | 2 | 3 |

9 | 10 | 11 |

17 | 18 | 19 |

Now I bring all the terms together to get an equation.

T = N – 17+N – 16+N – 15+N – 7+N

= 5N – 56

I then check if the equation is correct.

N = 18

T = (5x18) – 56

= 34

We find that this new equation work in this case, but try it with a different t-number to confirm its validity.

N = 50

T = (5x50) –56

= 194

Again, the equation has produced another correct result. I try it again one more time to make sure.

N = 22

T = (5x22) – 56

= 54

So I have produced another equation that works with in this grid, but would it work in the next grid size?

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 |

Conclusion

T = 5N – 63

63 / 7 = 9

So now what I can say is to figure out the t-total of a t-shape in any size grid, you need the formula “T = 5N – X”. The way you find out the t-total is as follows.

You firstly choose the t-shape you wish to find the t-total for. You then look at the grid size e.g. 7 by 7, and take the “7” from the grid size and multiply it by 7. This is your “X” value. You then multiply the t0number of the t-shape you are working out the t-total for and subtract the X value away from it. You should see, if done correctly, that the result would correspond with that of the real t-total.

What you can also figure out with this method is the t-total of an upside-down t-shape. To do this, you do as above, but instead of subtracting the X value away from 5N, you add it on. Note that all of the above equations have been based on upright t-shapes and don’t work if the t-shape is on its side.

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

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