If I substitute my T Numbers for N and the other numbers in the squares in relation to N I should be able to find the formula.
E.g.
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Change this Blue square to this red Square:
N-19
N-18
N-17
N-9
N
Then if you add up all the N’s and the numbers our formula for the T total would be
T=5N-63
To make sure the Formula is correct I will test it:
1+2+3+11+20=37
T=37 T also=5n-63
5n=100 100-63=37 This proves that the formula is correct
Key
T= T Total
N=T Number
G=Grid Size
The Formula T=5n-63 works for any T Shape on a 9*9 grid but now I have to see if it works on any other grid size.
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So if this is to have the same formula then the T Total would be
T=5N-63
T=130-63
T=67
To see if this is correct to that formula I would try and work it out by just adding the numbers together.
T=7+8+9+13+18
T=55
So This means that the formula T=5N-63 only works on a 9*9 grid and that I will need to find a different formula for the 5*5 grid
Again I will substitute my T Number with N and then change the other numbers in relation with N so I can find the formula.
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Change this Yellow T To This Red T
N-11
N-10
N-9
N-5
N
If I add up all the N T I end up with the formaula t=5n-35
So to check this with our original answer of just adding up the answer of the formula so
T=5n-35 5n=90
90-35=55
T=7+8=9+13+18
T=55
So that proves that 5n-35 is correct on a 5*5 grid
Now I Will test out a 4*4 grid to make sure that every grid has a different formula.
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To find the formula I will again need to substitute the T number with N and the other numbers in relation with N
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change this blue T into this Green T
N-9
N-8
N-7
N-4
N
So if you add up all the N T Shape we get the formula T=5N-28
So Now if I add up the original T shape I get:
T=6+7+8+11+15
T=47
Now I need to test if the formula gives me the same answer
T=5n-28 5n=75
T=75-28
T=47 So that proves that 5n-28 is the correct formula for a 4*4 grid.
Now I Will put my answers so far into a table to see if I can find anything else to help me
Size Of Grid: 9*9 5*5 4*4
………………………………………………………………
Formula: T=5n-63 T=5n-35 T=5n-28
I can see that there is a relationship between the formula and the size of grid. I have noticed that for every formula you take a multiple of 7 and that if you divide that number by the size of the grid you get the answer 7 so to get the answer for any grid size you have to multiply the grid size by 7 and take it from 5n
E.g.
N-2g+1
N-2g
N-2g-1
N-g
N
Add All These up To Get The formula T=5N-7g
E.g.2
If The grid Size is 5*5 you multiply 5 from the grid size and 7 to give you 35 you then take that away from your T Number multiplied 5 times so the formula would be
T=5N-7g
and this would be the formula for any grid size.
Key
T= T Total
N=T Number
G=Grid Size
If This formula was correct then on a 6*6 grid the formula would be: T=5N-42
So That would mean that the answer for the T Total should be
5n=145
145-42=103
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T=16+17+18+23+29=103
So That proves that the formula T=5N-7g is correct for every grid size
Now I Am going to translate the T Shape to another position on a 9*9 grid by using vectors
I have chosen 2 random T Shapes and the vector between them is{6}
{1}
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I can see that if we add 6 from one of our numbers on the blue T Shape and take away 9 we get the corresponding number on the red T shape.
If I put that into a Formula: T=N=6-9
I can see from the formula that both parts of the vector are in the formula, the six that you and the one * grid size.
If we substitute the vectors {6} into{A} I will now be able to form an algebraic
{1} {B}
formula. The formula would then become T=N=A-Bg
E.g.
26=29+6-9
26=29+[a]-[1g])
Key
T= T Total
N=T Number
G=Grid Size
Now I have to find the formulas for T after they have been rotated
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N-2G-1
N-2G
N-2G+1
N-9
N
Add This up To get T=5n-7g
N-G+2
N
N+1
N+2
N+G+2
Add This up To Get T=5n+7
N
N+G
N+2G-1
N+2G
N+2G+1
Add This up to Get T=5n+7g
N-G-2
N-2
N-1
N
N+G-2
Add This up To get T=5n-7
Position Of T Shape
Formula For T Total
Upright
T=5n-7g
90 Clockwise
T=5n+7
180
T=5n+7g
90 Anti clockwise
T=5n-7
Note that 90 Degrees Rotations do not include ‘g’ the grid size so these formulae are independent of grid size.
I will now Rotate a T shape 180 Degrees about an external point using the vectors {2}
{-1}
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I have noticed that if u double the vector u reach the corresponding T Number straight away and if we use x and y instead of numbers we will now have 2*{ x }
{-yg}
So if we go 2x across from The T Number 30 we get to 34 and go down 2*yg
we take away two grid sizes which is equal to 18 which is also equal to going down 2 squares so we get the formula:
T=5N+7G+5(2x-2YG)
I will now apply a combined transformation so all I need to do is find the formula for the second transformation as I have the formula for a 180 degrees rotation about an external point: T=5N+7G+5(2A-2BG)
So If I Now translate by a vector {s}
{t} the formula will also include the translation {s}
{-gt} for each T Number , therefore the total; formula for the T-Total will be:
T=5n+7g+5(2x-2YG+S-GT)
Where T=T Total
N= T Number
G=Grid Size
{x} ……… is vector for point of rotation
{y}
{s}…………..is Translation Vector
{t}