# T-Total and the T-Number.

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Introduction

## Maths Investigation

CODE – tn = T-Number. tt = T-Total

Aim: I) Investigate the relationship between the T-Total and the T-Number

II) Investigate the relationship between the T-Total, the T-Numbers and the grid size, using translation

III) Investigate the relationship between the T-Total, the T-Numbers, the grid size and the transformations.

## Part I

We were working with a 9x9 grid, and we had to construct a t-shape like 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 |

And so on…

To make the T-TOTAL, we had to add all the numbers inside the T together, and the T-NUMBER was the number at the bottom of the T (as highlighted inside the T).

Middle

22 = 47

23 = 52

24 = 57

25 = 62

Just from these results I can see that the T-Total rises by 5 every time the T-Number rises by 1. This means that there is a 1:5 ratio in terms of the T-Total and T-Number.

After this I had to work out a formula showing the relationship between the T-Total and the T-Number. To start with I knew it would be 5 x the T-Number, as the tt goes up 5 every time the tn goes up 1. Also, if I added the amounts that the numbers inside the T are subtracted from the tn, then it could finish the formula.

Therefore I did the following sum, 19 + 18 + 17 + 9 = 63.

As they were subtracted from the tn, it means that they

are all minus, my final formula is 5n – 63. As far as I know this will only work

for the 9 x 9 grid as the numbers in the T-Shape will differ in different grid sizes.

I will now test the formula to check it works for the 9 x 9 grid.

20 x 5 = 100-63 = 37 CORRECT

24 x 5 = 120-63 = 57 CORRECT

Therefore the correct formula for the 9 x 9 grid is “(5 x T-Number) – 63”.

Conclusion

For the T, and upside down T, the formula is 5n +/- (7 x Grid Number). This is due to a couple of factors. First of all it is 5 x T-Number (like in all of the equations) because there is 5 numbers in the T. Then it is +/- 63 (in the case of the 9 grid) because if you work out the sum of the numbers that are taken away (or added for the upside down T) from the T-Number that it will add up to +/- 63, and it will change for different grid sizes, meaning the number is different.

9 x 9 7 x 7

For the sideways T’s, it is 5n +/- 7. It is + for the T turned 90* clockwise, and – for the anti-clockwise. When added, each T-Number’s answer is 14 apart. After looking into it, the clockwise is always +7, and anti-clockwise –7 from the answer of 5 x the T-Number. This works for all grid sizes, because unlike T, and upside down T; there is one T-Total for each T-Number across all grid sizes when done sideways.

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

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