# T-Totals (A*) Firstly I have chosen to look at the 9 by 9 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five

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Introduction

T-totals

Introduction:

During this investigation I will be looking at different sized number grids, for example: (9 by 9, 10 by 10, etc…..). By looking at these different grids I will be drawing shapes on them called the t-shape. When I have drawn these, I will then be adding up the five numbers inside the t-shape. The sum of these five numbers will be called the t-total. Once I have found the t-total I will then record the number at the base of the t-shape which will be called the t-number. After recording all these figures I will then try and find whether there is a relationship between the t-number and the t-total thus coming up with a formula that links them.

Grid 1: (9 by 9)

Firstly I have chosen to look at the 9 by 9 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five and then I will be moving onto the second grid.

Number 1:

11 | 12 | 13 |

21 | ||

30 |

The t-number in this case will be 30.

The t-total is 30+21+13+12+11 which will give us 87.

T-number: 30

T-total: 87

Number 2:

12 | 13 | 14 |

22 | ||

31 |

The t-number in this case will be 31.

The t-total is 31+22+12+13+14 which will

give us 92.

T-number: 31

T-total: 92

13 | 14 | 15 |

23 | ||

32 |

Number 3:

Middle

48

The t-number in this case will be 48.

The t-total is 48+38+28+27+29 which will

give us 170.

T-number: 48

T-total: 170

Formula:

After investigating the t-numbers from 44 to 48 and comparing them with their t-totals, I have noticed that every time I increase the t-number by one the t-total goes up by five. I found this out by doing the table seen below:

Nth term | 1 | 2 | 3 | 4 | 5 | |||

T-number | 44 | 45 | 46 | 47 | 48 | |||

T-total | 150 | 155 | 160 | 165 | 170 | |||

Difference | +5 | +5 | +5 | +5 |

This table shows us that 5 multiplied by the nth term added to 145 will give us the t-total.

Overall formula:

I have also figured out a formula that will give us the t-total of any chosen t-number at random on a 10 by 10 grid. I did this by calling the t-number the nth term in each case and working out a formula from there, what I have done can be seen looking at the table below:

T-number* | 44 | 45 | 46 | 47 | 48 | |||

T-total | 150 | 155 | 160 | 165 | 170 | |||

Difference | +5 | +5 | +5 | +5 |

*nth term

Formula: 5n-70.

Checking:

I will now be checking if the formula I have come up with was correct, I will do this by choosing a random number on the 10 by 10 grid and substituting the t-number into the equation and coming up with the t-total without adding up the rest of the numbers. I will then add up the t-total and if it matches my prediction my formula will be correct.

The random number I have chosen is 72; this will be the t-number for checking my formula. I am now going to substitute the number into the equation:

T-total = 5(72)-70

T-total = 290

Now that I have got my prediction I will do what I have done for all the other numbers and find it out manually.

Predicted number:

51 | 52 | 53 |

62 | ||

72 |

The t-number in this case will be 72.

The t-total is 72+62+52+51+53 which will

give us 290.

T-number: 72

T-total: 290

My formula has turned out to be correct, I will keep this information and move on to the third grid.

Grid 3: (11 by 11)

Thirdly I have chosen to look at the 11 by 11 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five and then I will be moving onto the final grid.

Number 1:

60 | 61 | 62 |

72 | ||

83 |

The t-number in this case will be 83.

The t-total is 83+72+61+62+60 which will give us 338.

T-number: 83

T-total: 338

Number 2:

61 | 62 | 63 |

73 | ||

84 |

The t-number in this case will be 84.

The t-total is 84+73+62+63+61 which will

give us 343.

T-number: 84

T-total: 343

62 | 63 | 64 |

74 | ||

85 |

Number 3:

The t-number in this case will be 85.

The t-total is 85+74+63+64+62 which will

give us 348.

T-number: 85

T-total: 348

Number 4:

63 | 64 | 65 |

75 | ||

86 |

The t-number in this case will be 86.

The t-total is 86+75+64+65+63 which will

give us 353.

T-number: 86

T-total: 353

Number 5:

64 | 65 | 66 |

76 | ||

87 |

The t-number in this case will be 87.

The t-total is 87+76+65+66+64 which will

give us 358.

T-number: 87

T-total: 358

Formula:

After investigating the t-numbers from 83 to 87 and comparing them with their t-totals, I have noticed that every time I increase the t-number by one the t-total goes up by five. I found this out by doing the table seen below:

Nth term | 1 | 2 | 3 | 4 | 5 | |||

T-number | 83 | 84 | 85 | 86 | 87 | |||

T-total | 338 | 343 | 348 | 353 | 358 | |||

Difference | +5 | +5 | +5 | +5 |

This table shows us that 5 multiplied by the nth term added to 333 will give us the t-total.

Overall formula:

I have also figured out a formula that will give us the t-total of any chosen t-number at random on a 11 by 11 grid. I did this by calling the t-number the nth term in each case and working out a formula from there, what I have done can be seen looking at the table below:

T-number* | 83 | 84 | 85 | 86 | 87 | |||

T-total | 338 | 343 | 348 | 353 | 358 | |||

Difference | +5 | +5 | +5 | +5 |

Conclusion

I also noticed that the number taken away from 5n is the same as seven multiplied by the grid number every time, so 9 would be 63, 15 would be 105 and so on.

By figuring this out I managed to make a formula linking the t-number of any grid to its t-total using a quick simple formula. The formula is as follows:

T-total = 5n-7g

N=the t-number

G= the grid number

Now I will simply try the formula out on a random t-number chosen on a random grid and see if my formula works out correctly.

The random t-number I have chosen is 109 from the grid 11 by 11. I am now going to substitute these figures into the equations I have got and predict the t-total without adding the numbers. I will then check my prediction and if it matches the answer I add up I will know that my formula is correct.

Prediction:

T-total = 5n-7g (n=109 , g=11)

(5*109) - (7*11 545-77

My prediction is that the t-total for t-number 109 is going to be 468. I will now check to see if my answer is correct.

Check formula:

86 | 87 | 88 |

98 | ||

109 |

The t-number in this case will be 109.

The t-total is 109+98+87+86+88 which will

give us 468.

T-number: 109

T-total: 468

My formula is correct; I have solved the relationship between t-numbers and t-totals on any sized grid. The formula (5n-7g) is the answer to this problem.

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

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