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
In part III, I will rotate the T in different grid sizes and investigate any relationships and formulas.
Firstly, I tried an upside-down T, with the top number being the T-Number.
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.
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.