Every time you move up a line e.g. from T to D you tack away the grid size.
So the formula will work with algebra because when we relate the t-number to the number above it you just tack away the grid number.
The algebra formula will be 5N-7grid-1+1
Simplyied would be 5N-7g
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90˚ rotation
7x7grid
Using a basic t-shape I am trying to find a formula to get the t-total after turning it 90˚ clockwise
I will now relate each cell to the t-number (t)
A=t-2g-1
B=t-2g
C=t-2g+1
D=t-g
E=t+1
F=t+2
G=t+2+g
H=t+2-g
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When you put this all together into a formula you get
5N-7g for the t-shape before the rotation
5N+1+2+2+2-g+g simplyied would be 5n+7
This formula will work on any size grid and any t-number for e.g.
6x6grid
The t-number is 27 so using the formula the t-total would be 5x27-7x6=93 14+15+16+21+27=93
After the 90˚ turn clockwise the t-total would be
5x27+7=142 27+28+29+23+35=142
So the formula works on different size grids and different t-numbers
7
180˚ rotation
Using a basic t-shape I am trying to find a formula to get the t-total after turning it 180˚ clockwise
I will now relate each cell to the t-number (t)
A=t-2g-1 E=t+g
B=t-2g F=t+2g
C=t-2g+1 G=t+2g-1
D=t-g H=t+2g+1
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When you put this all together into a formula you get
5N-7g for the t-shape before the rotation
5N+g+2g+2g+2g-1+1 simplified would be 5n+7g
This formula will work on any size grid and any t-number for e.g.
7x7grid
The t-number is 17 so using the formula the t-total would be 5x17-7g=26 2+3+4+10+17=26
After the 180˚ turn clockwise the t-total would be
5x17+7g=134 17+24+30+31+32=134
So the formula works on different size grids and different t-numbers
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270˚ rotation
6x6grid
Using a basic t-shape I am trying to find a formula to get the t-total after turning it 270˚ clockwise
I will now relate each cell to the t-number (t)
A=t-2g-1
B=t-2g
C=t-2g+1
D=t-g
E=t-1
F=t-2
G=t-2-g
H=t-2+g
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When you put this all together into a formula you get
5N-7g for the t-shape before the rotation
5N-1-2-2-2-g+g simplifies would be 5n-7
This formula will work on any size grid and any t-number for e.g.
8x8grid
The t-number is 46 so using the formula the t-total would be 5x48-7g=184 29+30+31+38+46=184
After the 270˚ turn clockwise the t-total would be
5x46-7=233 46+45+44+36+52=233
So the formula works on different size grids and different t-numbers
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Translations
Translations are were we can move the t-shape around the grid but before we can do this we have to create a formula for the movements.
8x8grid
To move the t-shape across to the right 1 cell you add 1 onto each square so you add 5 to the t-total
So to move the t-shape 3 cells across to the right you need to add 15 to the t-total
But when you move to the left you tack 1 from each cell so you take 5 off the t-total
So to move the t-shape 3 cells across to the left you need to take 15 off the t-total
This means that all movement on the x is equal to +or- 5 so you times the grid movement eg. +1, -5, +8 by 5
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X can be a – number and a + number
- If you won’t to go left
+ If u won’t to go right
So -5 means move across 5 cells to the left
And +3 means move 3 cells to the right.
And the same for the Y but we need the formulas before we can put the numbers in the coordinates.
6x6grid
When you move down a cell you add the number of the grid e.g. 2 to 8 = 6 15 to 21 = 6 3 to 9 = 6
To move the t-shape down1 cell you add the grid size onto each square so you add 5g to the t-total
So to move the t-shape 3 cells down you need to add 15g to the t-total
But when you move up you tack 1 grid number from each cell so u take 5g off the t-total
So to move the t-shape 3 cells up you need to take 15g off the t-total
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We can put this into a coordinates and make a formula
X
Y
X = the movement across
Y = the vertical movement
The equation given to this is
5N+7G+5X+5GY
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Transformations
I will use the info I’ve found to make a formula that covers all the ways of moving and turning a t-shape
9x9
I am going to move the t shape with 30 as its t-number +3 across and -3 down an turning it 90’ clockwise
The formula I will use is 5n+7+5x-5gy the 5n+7 is from the 90˚ rotation and the 5x is the horizontal movement and the 5gy is vertical movement.
5n+7+5x-5gy
Simplified; (5 x t-number) + 7 + (5 x +3) – ((5 x 9) x -3)
Using this formula the answer is 307
60+61+62+53+71=307
So the formula is right but now we need to change it for the 180˚ rotation
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I will now test the formula for the 180˚ turn and translation
9x9
I am going to move the t shape with 62 as its t-number -6 across and +6 up an turning it 180’ clockwise
The formula I will use is 5n+7g+5x-5gy the 5n+7g is from the 180˚!!" rotation and the 5x is the horizontal movement and the 5gy is vertical movement.
5n+7g+5x-5gy
(5 x t-number) + (7 x grid) + (5 x -6) – ((5 x 9) x +6)
Using this formula the answer is 73
2+11+19+20+21=73
So the formula is right but now we need to change it for the 270˚ rotation
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I will now test the formula for the 270˚ turn and translation
9x9
I am going to move the t shape with 66 as its t-number +6 across and +6 up an turning it 180’ clockwise
The formula I will use is 5n-7+5x-5gy the 5n-7 is from the 270˚ rotation and the 5x is the horizontal movement and the 5gy is vertical movement.
5n-7+5x-5gy
(5 x t-number) -7 + (5 x +6) – ((5 x 9) x +6)
Using this formula the answer is 83
7+25+16+17+18= 83
So the all the formulas are right and produce a table of results for all I have found
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Conclusion
This table is all the formals you need to rotate a t-shape
This table is for the movement or translation for a t-shape
5N-7g+5X+5gY
Use X+Y as coronets
9x9
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X movement is +4 Y movement is +1
5N-7g+54+5g1
T-total = 5N-7g+54+5g1
5x21-7x9+5x4+5x9
105-63+20+45
42+65
107
34+25+15+16+17=107
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