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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

3

4

5

6

7

8

9image00.pngimage00.png

10

11

12

13

14

15

16

17

18image00.png

19

20

21

22

23

24

25

26

27image01.png

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).

...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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