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Investigating the Relationship Between the T-Total and the T-Number

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Investigating the Relationship Between the T-Total and the T-Number

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

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5

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7

8

[IMAGE][IMAGE][IMAGE]9

10

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[IMAGE]18

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[IMAGE]27

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37

38

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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).

To start with I added all the numbers together inside the T, and for
20 the T-Total was 37 - (1+2+3+11+20 = 37). I then did 21,22, 23 and
so on up to 26. My results were as follows:

20 = 37

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

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T-number         Total         Relationship         Added         Inside         Finish         Numbers         Amounts         Grid         Rises        




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.

[IMAGE]


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".


Part II

In part II, I changed the grid size around to see if could find any
relationship between the formula or the 9 x 9 grid and any other grid
sizes.

I tried grid sizes from 4 x 4 - 8 x 8. Here are the results I came up
with:

4 x 4

10 = 22

11 = 27

12 = 32

13 = 37

5 x 5

12 = 25

13 = 30

14 = 35

15 = 40

6 x 6

14 = 28

15 = 33

16 = 38

17 = 42

7 x 7

16 = 31

17 = 37

18 = 42

19 = 47

8 x 8

18 = 34

19 = 39

20 = 44

21 = 49

Straightaway I realise that the 9 x 9 formula doesn't work, as the
resulting number would be too small. Instead I can times a number from
each grid by 5 to work out the formula.

50 - 22 = 28

60 - 25 = 35

70 - 28 = 42

80 - 31 = 49

90 - 34 = 56

100 - 37 = 63

From this, I can see that there is a gap of 7 each time, and the final
number is actually the product of the grid size x 7.

Therefore, I can say that the formula for the T-Shape in all grid
sizes is

"(T-Number x 5) - (Grid Size - 7)"

Part III

[IMAGE]In part III, I will rotate the T in different grid sizes and
investigate any relationships and formulas.

[IMAGE]




Firstly, I tried an upside-down T, with the top number being the
T-Number.

[IMAGE][IMAGE][IMAGE]

I worked out the tt for numbers 21-24 to see if I could find any
connection…

21 = 168

22 = 173

23 = 178

24 = 183

As I couldn't find any direct connection from the start, I decided to
add the numbers in the T together again. The total was 63, but this
time it was plus, rather than minus, which explains the big total.
Therefore, the formula for an upside down T is the same as an upright,
except it is plus (7 x grid number)…(5 x T-Number) x (7 x grid
number).

I now tried translating the T sideways. First, I turned it 90*
clockwise, then 90* anti-clockwise. These are my results:

CLOCKWISE ANTI-CLOCKWISE

21 =112 21 = 98

22 = 117 22 = 103

23 = 122 23 = 107

24 = 127 24 = 113

One noticeable thing is that they are 14 apart, meaning that AC will
be -7 and C will be +7. As they both go by five each time, it means
that the formula will start "5 x T-Number". There fore the final
formula for C is 5tn + 7, and for AC it's 5tn - 7.

I now need to test to see if it changes with each grid size, like the
formula for T and upside down T.

8 x 8

CLOCKWISE…10 = 57correct 14 = 77correct

ANTI-CLOCKWISE…11 = 48correct 12 = 53correct

7 x 7

CLOCKWISE…10 = 57correct 13 = 72correct

ANTI-CLOCKWISE…10 = 43correct 12 = 53correct

From all my results I can say that the formula for C and AC works for
all grid sizes.

Conclusion

In conclusion, I have found that first of all for every T-Shape there
is a ratio of 1:5, because there is 5 numbers in the T, and when they
are moved one space to the right, they move up 1, meaning when the
T-Number goes up 1, the T-Total will go up 5.

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.

[IMAGE]

[IMAGE]




9 x 9 7 x 7

N

N




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.



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