# T-Totals.I will investigate what makes up a T-total and what makes up a T-number. This will be done by drawing out a nine by nine grid first. Then looking at the grid I will try to see what kind of relationships there are

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Introduction

Adam May

T-Totals

What I will do:

I will investigate what makes up a T-total and what makes up a T-number. This will be done by drawing out a nine by nine grid first. Then looking at the grid I will try to see what kind of relationships there are. Then I will do the same again but with other sized grids e.g. a ten by ten grid. This will then show me if there any changes and if so what are they. I will look at these by drawing out the grids then, using a piece of paper I will cut out a T shape that will measure three squares across and three down. This will then allow me to clearly see the five numbers in question, with out getting them mixed up with other numbers. Then once I have taken a reasonable sample with the three square “T” I will increase its size to a five by five “T” shape and then this will show up any other possible connections between the T shape and the T number. I will also look at the “T” number, this is the number at the bottom of the “T” shape. It is always the largest number.

Example of what the cut out would look like;

Piece of paper

## Holes cut out

Middle

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

T shape 4 T shape 5

Again I have labelled these equations 4 and 5.

In T shape 4 is has shown me that the “T” number is 39, then I added up all the numbers in red in t shape 4 to give a “T” total of 117. This is because 1+2+3+4+5+12+21+30+39=117.

In T shape 5, 61 is the “T” number and the “T” total is 315 this is because 23+24+25+26+27+34+43+52+61= 315

I have looked at both of these results very hard and come up with an idea. My idea is that If for example you look at T shape 1 the “T” number is 20, but if you replaced this number with “N” you get;

“T” number = N

Then the number in the “T” square, directly above the “T” number is then nine places back in the grid so it is N – 9. The number directly above that is then N-9-9 = N-18. The two remaining numbers in the “T” shape are N-18-1 and N-18+1.

This also works with T shape 4 but some alterations have to happen because it is a five by five “T” shape.

If we take the “T” number as 39 this then transforms in to N. This means the number directly above that N-9, the one above that is N-9-9= N-18, the one above that is N-9-9-9= N-27 and the one above that is N-9-9-9-9= N=36 the other number would be N-36-2, N-36-1,N-36+1 and N-36+2.

These formulas would help if only the “T” number was given and I was asked to find out what the “T” shape is, but looking in to it further I have realised it will help me to work out the “T” total.

Then as I was looking at this, I noticed a what was similarity between T shape 1,2 and 3 when I used the formula N+(N-9)+(N-18)+(N-18-1)+(N-18+1) = 5N-63 and doing this you end up with the “T” number.

For example look at “T” shape 2 and use the above formula:

N+(N-9)+(N-18)+(N-18-1)+(N-18+1)= 5N-63 which equals187 which is the “T” total for “T” shape 2.

So now I have found a formula for the five square “T” shape and checked it by making sure it works with “T” shape 2. I now I must change this formula To suit the nine square “T” shape.

I looked at this problem and worked out the change for the formula. This is the formula for a nine square “T” shape

N=(N-9)+(N-18)+(N-27)+(N-36)+(N-36-1-1)+(N-36-1)+

(N-36+1)+(N-36+1+1)=9N-234 then I checked this by using “T” shape five and the equation is correct because it gave me the answer of 315 which is the “T” total.

This has proved that I have now found the formula to work out the “T” total just from the “T” number, or the “T” number just from the “T” total.

I will now investigate a seven by seven “T” shape just to make sure that the formula works, I will also change the size of the grid to see if this affects the formula in any way. I will change the grid to a ten by ten 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 | 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 |

Conclusion

I will check my working one more time to make sure I’m correct.

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 |

“T” shape 9

I have labelled the “T” shape, “T” shape 9. I have put it on a eight by eight grid. It is a five by five “T” shape.

I will change the formula to incorporate the size of the grid.

N+(N-8)+(N-16)+(N-16-1)+(N-16+)= 5N-56

When I work out this equation I get the answer 119, which proves my theory, is correct because 119 is the “T” total.

Summery:

So in this I have shown that I have created a formula that will allow you to work out any “T” number or “T” total. This can simply be done by adjusting the formula to suit the problem. I have found this formula will solve any questions whatever the size of “T” shape or grid.

Evaluation:

I feel that I have worked extremely hard in this coursework to find out my formula. I am now confident that whatever size grid I am given I could adjust the formula to suite, because I have explored any of the possibilities. I am also confident whatever size “T” shape I am given I would also be able to solve this using my formula.

I liked this question because it tested me, and it was a challenge to find the answer.

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

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