As I said before there are 5’n’ and the sum of these numbers =
N-19+n-20+n-21+n-9+n = 5n- 63
This has then proved that my prediction was correct.
Now I am going to investigate the relationship between the T-Number (n) and T-total in any size grid.
8*8 Sized Grid
After some more results I then tabulated them and found anther relation.
Again you can see that the T-Total increase by 5. This therefore means I will do the same as before and look at the difference between 5n and the T-Total
Now from this the difference is now 56 and therefore, for an 8 by 8 grid I predict that the formula will be 5n – 56. As before, I will again use ‘n’ rather than a number.
10 * 10 Grid
As I have already shown in the previous grid through calculation, I have determined that 5n is the constant difference so that to repeat a table showing data from the T- total against the T- Number will be pointless.
From the above table of results I can tell that the difference is 70 therefore for a 10 y 10 grids the formula will be 5n – 70. Again to prove this I will use ‘n’.
Now for the three grids that I have already done I will tabulate the formulas.
Now, by just looking at the differences and relations in the numbers I can see a pattern and that it will always be the formula 5n – 7 * Grid Width (G).
Now I will combine both ‘n’ and ‘G’ into one formula.
T-Total = 5n – 7G
Now I will look at different transformations
Translation
Now I will look at the relationship between the T- Number and T-Total as the T shape is moved onto different vectors on grid.
9 * 9 Grid Translations
I will now translate the T- shape to the vector
With the above formula I can predict that the general formula for vectors is
N + A –BG
As I did more example of vector translating I found that horizontally the number change increased only by one each time. I then found out that as the shape moved vertically, the numbers changed by 9 each time, so I called IT ‘BG’.
I then, to prove my prediction used the terms in a T-shape.
9 * 9 Grid Reflections
I will now investigate the relationship between the T number and T total when reflected from one position on the grid to the other.
I will again start on a 9 * 9 grid
Image 1 = T- Number =21 + 3 T-Total = 42 +15
Reflect 1= T-Number=24 T=-Total=57
Image 2= T- Number= 65 T-Total=262 +25
Reflect2= T-Number= 50 +5 T-Total=287
From these figures i can see a relationship emerging although only for the vertical reflection. For every square the shape has moved each one has caused the number to gain 5 to the T-Total.
To prove that this is true, I will label a T shape and inside it put the term ‘S’. This term stands for the number of squares the T-shape has moved.
With all my previous workings and examples I can strongly confirm my prediction and state that ‘-63’ can be replaced with ‘7G’.
No I will look at the Horizontal line of reflection.
Image 1- T-Number= 29 +9 T-Total =82 +171
Reflect 1-T-Number=38 T-Total =253
Image 2 – T-Number=25 +27 T-Total=62 +261
Reflect 2 – T-Number=53 T-Total=323
As in the vertical reflection another pattern has emerged and this time for each square that the shape moves down 9 is gained onto the T-Total. With the relationships compiled from previous calculations I can also work out the T-Total.
The formula for a normal T-shape on a 9*9 grid is 5N – 63 then the formula for a reflected t shape can be worked out and adapted by just flipping it over.
It is from this depiction that we see that the formula is 5n + 63. Now to combine both together.
5N +5(G5) +63. Like before, I will change the ‘+63’to ‘+7G’. I WILL NOW USE ‘Reflect 1 and 2@ to check my formula.
Checking:
T-Total = 52 +61 +69 + 70
= 323
Formula =5(N+GS) +7G
=5(25+ (9*3) +63
= (5*52) +63
=260+63
=323
9 * 9 Grid Rotations
Finally I am going to look at the relationship between the T-Total and T-Number as they are rotated clockwise up to 270 degrees on a 9*9 grid.
The formula for a 90 degrees=5N +7
The formula or 180 degrees= 5N+7G
The formula for 270 degrees= 5N-7
