I will now try to work out a relationship/formula for the t-shape when it is rotated to a different position, I will begin with a t-shape upside down. I will begin with T-Shapes on a 9 wide grid.
Whilst looking at the structure, it is similar to the one of an upright T other than the values are positive as you can see on the right.
Hopefully the formula will equal T-Total = T+T+T+T+T+g+g+g+g+g+g+g (5t+7g) as this is the case when looking at the structure and comparing to the previous formula.
I will test this on a 9 wide grid.
T-Total = 88
T-Number = 5
Using formula T-Total = 5x5+ (7x9) = 88
So the formula works for t-shapes rotated upside down, I will now try it on a selection on different sized grids to see if it still holds.
8 wide grid
T-Total = 66
T-Number = 2
Using formula T-Total = 5x2+ (7x8)
= 66
The formula works in this case; I will now test it on a 6 wide grid.
6 wide grid
T-Total = 57
T-Number = 3
Using formula T-Total = 5x3+(7x6)
= 57
The formula is again correct, so the correct formula for t-shapes rotated upside down is:
I will now begin working out a relationship/formula for a t-shape rotated on its side as shown on the right.
I began by placing the t-shapes on a 9 wide grid and gathered some results.
I will look at the structure of the t-shape as shown on the right to come up with a basic formula.
Using the t-structure on the right I can work out the t-total.
T-Total = T+T+1+T+2+T+2-g+T+2+g
= 5t+7
When this formula is tested on all the t-shapes above it works.
I will now see if my formula works on different sized grids.
8 wide grid
T-total = 52
T-number = 9
Using formula T-Total = 5x9+7
7 wide grid = 52
T-Total = 47
T-Number = 8
Using formula T-Total = 5x8+7
= 47
So the formula also holds for different sized grids.
I will now begin working out a relationship/formula for a t-shape rotated on its side as shown on the right.
I began by placing the t-shapes on a 9 wide grid and gathered some results.
Before getting to the structure of the rotated t-shape I believe the t-total will be 5t-7 as the case was before that when it was rotated 180 degrees the values became positive or negative.
Here is the t-structure for the rotated t-shape.
So the T-Total = T-2-g+T-2+T-1+T=T-2+g
= 5t – 7
This is as I expected. I will test this formula to prove it works.
I will test this on a 9 wide grid.
T-Total = 58
T-Number = 13
Using formula T-Total = 5x13-7
= 58
The formula works, I will test it again.
T-Total = 63
T-Number = 14
Using formula T-Total = 5x14-7
= 63
So the formula which works for t-number rotated 90 degrees anticlockwise is:
The formula also holds whilst been tested on different sized grids.
The general formulas for all the t-shapes when on any size grid are:
When the t-shape is normal – T-Total = 5t-7g
When the t-shape is rotated 180 degrees – T-Total = 5n+7g
When the t-shape is rotated 90 degrees clockwise – T-Total = 5t+7
When the t-shape is rotated 90 degrees anticlockwise – T-Total = 5t-7
I will now begin to translate the t-shape onto grids using different vectors.
I shall start by using with a = 4.
This is the t-structure of the original T above. Using the vector of 4/0 each number increases by 4 which is “a”. I will add the 4 to the structure to make a formula for the new t-total.
So the t-total using a vector of is:
New T-Total = n-2g-1+a+n-2g+a+n+2g+1+a+n-g+a+n+a
= 5n + 5a – 7g
= 5 (n+a) – 7g
This also applies when a is a negative number.
To work out the new T-Number the formula is simply. New T Number = n+a
I will now try translating the T-Shape onto a grid using a vector of .
To begin with I will make b=-4 .
Using this vector each number from the old t-shape increases by 36 in the new one. To work the formula out I will add 36 to each part of the t-structure as when all combined the new t-total can be calculated.
So the new t-total = 5n-7g+180
But when you refer to the sizes of the grid the formula changes.
So the new t-total = 5n-7g-5(bxg)
I will test this formula with a vector b=-3.
Using formula T-Total = (5x20)-(7x9) +5(3x9)
= (100-63) + (5x27)
= 172
Once checked the answers correspond with the formula results meaning it works. SO the formula which works out the new t-total when b of the vector is:
The formula to work out the new t-number of the new t-shape is:
Both these formulas hold for negative values.
I will now try the vector. I will use my previous formulas to get an overall formula.
= 5 (n+a) – 7g
= 5n-7g-5(bxg)
So will equal
So the new t-total using vector is 5n-7g+5a-5bg. The 5bg part of the formula is from the 5(bxg) of the original ob vector shown above.
I will test this formula or a 9 wide grid with a vector of .
New T-Total = 5(20+2+3x9)-7x9
= 245-63
= 182
When put on a graph the answers correspond with that from the formula.
I will now try this formula on a 7 wide grid using the same vector as before.
New T-Total = 5(16+2+3x7)-7x7
= 5x39-49
= 195-49
= 146
When put on a graph the answers correspond with that from the formula so the formula is correct and works for grids of any width.
