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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

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4

5

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7

8

9image00.pngimage00.png

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18image00.png

19

20

21

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25

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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).

...read more.

Middle

21 = 42

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.

image12.pngimage10.pngimage11.pngimage09.png

image13.png

 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”.

...read more.

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.

image02.pngimage03.pngimage05.pngimage04.png

image07.pngimage06.png

9 x 9                                                                                            7 x 7

image08.png

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.

...read more.

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