(T-19) + (T-18) + (T-17) + (T-9) + (T)
= 5T – 63
Let T-number = T
Let T-total = n
Formula = 5T – 63
I tested that when:
T = 80
n = 337
337 = 5T – 63
337 = (5 x 80) – 63
= (5 x 80 = 400) – 63
= 400 -63
= 337
337 = 337
Formula works.
Below is a t-shape, and in each cell how the number is connected with T on a 9x9 grid
Check the formula:
(T-19) + (T-18) + (T-17) + (T-9) + (T)
= 5T – 63
So this is correct
10x10 Grid
(T-21) + (T-20) + (T-19) + (T-10) + (T)
= 5T-70
Let T-number = T
Let T-total = n
Formula = 5T – 70
I tested that when:
T = 88
n = 370
370 = 5T – 70
370 = (5 x 88) – 70
= (5 x 88 = 440) – 70
= 440 - 70
= 370
370 = 370
Formula works.
Below is a t-shape, and in each cell how the number is connected with T on a 10x10 grid
Check the formula:
(T-21) + (T-19) + (T-20) + (T-10) + (T)
=5T-70
So this is correct
After finding out the similarity of the 8 x 8, 9 x 9 and 10 x 10 grid, I am now going to work out a formula that should work on any grid size with a T-Shape.
Let GxG = Grid Size
Let G = Length/width of grid
Let T = T-Number
How did I get 5T-7G?
Here is how I found out the relationship between the T-Number, T-Total and Grid Size:
Here I have constructed a random GxG Grid to prove that my Formula of 5n-7G will work on any Grid size, and any position
15 x 15 Grid
32 Is my T-Number
T = 32
5T = 5x32 = 160
G = 15
7G = 7x15 = 105
I now have enough data to prove that my formula works
5T - 7G
160 – 105 = 55
T-Total = 1+2+3+17+32 = 55
So my Formula works because the T-Total = 55, and my Formula gives a T-Total for the T-Shape as 55.
9 x 9 Grid
T =59
5T = (5x59) = 295
G = 9
7G = (7x9) = 63
T-Total = (40+41+42+50+59) = 232
I will now apply my formula onto this T-Shape on the 9x9 Grid to prove it will work on any Grid size with a T-Shape at any position
5T-7G
(5x59)-(7x9)
= 295-63
= 232
Is equal to the T-Total, so my formula 5T-7G Works.
The relationship between the T-Total, T-Number and Grid Size are all summed up into the formula 5T-7G.
This formula can be used on any Grid Size, with the T-Shape at any position.
PART 3
I am going to investigate the relationship between the T-Total, the T-Numbers, the Grid Size, and different transformations.
I am going to use the transformation of translation on the following T-Shape to the right on this 10 x 10 grid
10x10 Grid
I will now work out a general formula for a T-Shape that has been translated to the right, for any grid size because I can see the connection with a translation of 2 to the right and the grid.
((T-2G)+1) + ((T-2G)+2) + ((T-2G)+3) + ((T-G)+2) + (T+2)
=5T-7G+10
I will now get a formula by drawing part of a GxG grid and using a T-shape that has been translated x times to the right.
Let x = number of times/squares translated to the right
I will now work out a general n that has been translated x times to the right
((T-2G)+x-1) + ((T-2G)+x) + ((T-2G)+x+1) + ((T-G)+x) + (T+x)
=5T-7G+5x
This formula should work out the T-Total for the translated shape that has moved 6 times to the right, It is the Lined shape.
T = 32
G = 15
X = 6
= (5x32)-(7x15)+(5x6)
= 160 – 105 + 30
= 85
Which is the T-Total for the Translated T-Shape, this proves the formula works.
I will also test the shape again to make sure the formula 5T-7G+5x works for any shape on the grid
I have once again used the formula on another translated T-Shape, this shape has been moved 8 times to the right, it is blank shape with the dark border.
T = 32
G = 15
X = 8
= (5x32)-(7x15)+(5x8)
= 95
This is the T-Total for the shape, which proves my formula works.
I will now test the Formula 5T-7G+5x on part of a 9x9 Grid to show it will work on any grid size.
T (original shape) = 20
G = 9
X = 5
=5T-7G+5x
= (5x20)-(7x9)+(5x5)
= 62
I am now sure the formula works on any Grid size, with any x places to the right.
I will now find out a formula to translate the T-Shape y places down on any grid.
Let y = Number of times/squares translated down
I will now do all the same things as I did translating the T shape across, but now I am going to Translate it down, which means I will be finding a general T-shape that has been moved y times down for any grid size and a general formula.
Then I will combine both Moving the T shape Across And moving it down, which will enable me to get a general formula for a general T shape that has been moved x times right and x times down.
10x10 grid
Let y = number of times/squares translated down.
I will now work out a general n that has been translated x times down.
T = 22
G = 10
Y = 2
5T-7G+50y
(5x22) - (7x10) + (50x2)
=110 – 70 + 100
= 140
This is the T-Total for the translated shape. This proves my formula works.
Also, I can prove it works by the following:
((T-2G)+10y-1) + ((T-2G)+10y) + ((T-2G)+10y+1) + ((T-G)+10y) + (T+10y)
= 5T-7G+50y
= My formula
I noticed that this formula only works on a 10x10 Grid, in order to make it work on any grid size, I have to change the formula to: 5T-7G+ ((5xG)xY)
This gives me the formula 5T-7G+((5xG)xY) as the y=((5xG)xY). This formula can be used on any shape, I will now test this formula to prove it.
I will now test this formula on the 10x10 grid as I tested the formula 5T-7G+50y
T = 22
G = 10
Y = 2
5T-7G+((5xG)xY)
(5x22) - (7x10) + ((5x10)x2)
=110 – 70 + 100
= 140
I will now prove my formula works on any grid by testing it on part of another grid, a 9x9 Grid.
T = 20
G = 9
Y = 6
5T-7G+((5xG)xY)
= (5x20) - (7x9) + ((5x9)x6)
=100 – 63 + 270
= 307
Which is the same as the T-Total of the translated shape 55+56+57+65+74=307
This proves my formula for Y will work on any Grid size with any translation of Y places down.
So far I have found a formula for X places to the right, and Y places down. I will now try and find a formula that relates both the X and Y to show a relationship between the T-Total, the T-Number, the Grid Size, and the Transformation (translation in this case)
Y-Grid X-Grid
X + Y on the grid
= 5T-7G+5x+((5G)xY)
T= 22
G= 10
X=4
Y=2
5T-7G+5x+((5xG)xY)
= (5x22) – (7x10) + (5x4) + (50x2)
=110 – 70 + 20 + 100
=160
Which is the T-Total for the t-shape on the 10x10 Grid
To make sure the formula works on any Grid I am going to test it on a 9x9 Grid.
T= 20
G= 9
X=3
Y=5
5T-7G+5x+((5xG)xY)
= (5x20) – (7x9) + (5x3) + (45x5)
=100 – 63 + 15 + 225
=277
Which is the T-Total for the t-shape on the 9x9 Grid
49+50+51+59+68=277
This proves that the formula 5T-7G+5x+((5xG)xY)
will work on any grid with any translation of x spaces to the right and y places down.
I have also realised that this formula ONLY works on right and down, in order for it to work on left and up you will need to change the values of “x” and “y” from +ve to –ve. This will give you the ability to go left and up. I also noticed that to move x places to the left and y places down, you need to change the “x” to –ve and leave the “y” as +ve. And vice versa.
THIS IS NOT NEEDED, JUST INCASE YOU WANT TO WORK ON ROTATION INSTEAD OF TRANSLATION, JUST THE BEGINNING OF IT.
I will now try and find the relationship between the T-Total, the T-Numbers, the Grid Size, and the rotation of the T-Shape.
I have rotated the T-Shape 90* Clockwise at point 32. I will now try and find the relationship between the T-Total, T-Number, Grid size and the rotation.
T=22, T-Total = 40
T=33, T-Total = 172
