T-shape 1
T-shape 2
Grid 2
T-shape1: Let n be 15
T number = n
T total= (n-9)+(n-8)+(n-7)+(n-4)+n
= 5n-28
T-shape2: Let n be 10
T number = n
T total= (n-9)+(n-8)+(n-7)+(n-4)+n
= 5n-28
The grid size is 4x4. Again, I get the same results moving the T-shapes to different positions on the grid. I find out that the size of the grid is a very important factor leading to this result. The formula is always equal to 5n-(7x grid size), which can be represented by 5n-7g.
The formula doesn’t work when…
Take a 9x9 grid as example
1) When the T number is less than 20 or when the T number is 1,10,19,28,37,46,55,64 and 73.
(Example is shown on the grid)
If the T number is 19, there will be no grid left on the left hand side of the T-shape and so that part will be stuck out. The same happen to all the numbers in the first row of the grid.
2) When the T number is less than 18
(Example is shown on the grid)
If the T number is 16, there is no grid left on the top part and so the top part will be stuck out.
Part 2 continued
T-shape1
T-shape2
Grid 1
T-shape 1: T-shape 2:
T number = 21 T number =35
Vector: (-2)
2
The difference between the two T numbers: 14
Grid size: 6
Hypothesis:
T number grid size(g) Vector Difference of T no.s
21 6 -2(a) 14
- 2(b)
Difference of T numbers= g(a)+b
Example: 6x2+2=14
I found out that you could never get this result if you multiply the grid size with the negative sign. Also, this negative sign doesn’t indicate anything but the action, moving down.
Part 2 continued
Translation 1
Translation 2
Grid 2
Example 1
T number Grid size Vector Difference of T no.s
54 10 (5)(a) 35
- -3 (b)
Now I use the formula g(a)+b
10x5+3=53
It doesn’t work but it does if I swap a and b around
10x3+5=35
For Translation, I found out that the relationship of between the 2 T numbers can be represented by g(a)+b or g(b)+a.
Example 2
T number Grid size Vector Difference of T no.s
49 10 (-6)(a) 44
- -5 (b)
The formula doesn’t work using
g(a)+b: g(a)+b:
∵10x6+5=65 ∵10x5+6=56
But it does work when I use this formula: g (b)-a
10x5-6=44
Unlike what I mentioned earlier, the negative sign of a is kept in this example. The reason is because both of the vectors have a negative sign, so you must keep one of the signs in order to get the desired result.
As a conclusion for this part, the formulas representing the relationship between the T-total, the T-numbers and the grid size are g(a)+b and g(b)+a.
Part 3
T-shape before rotation
T-shape after rotation
I turned the T-shape 180°and it is the reflection of the original T-shape. The formula must be the opposite of that of the original shape.
The formula for the original T-shape: 5n-49
The formula for the revered T-shape: 5n+49
T total of the reversed T-shape=5n+49
L.H.S=31+38+44+45+46
= 204
R.H.S=5n+49
=5x31+49
=204
∵L.H.S.=R.H.S.
∴The reverse in the minus sign has worked.
Part 3-Mirror image(180° rotation)
Now I try the formula, 5n+7g on this grid.
The formula for the original T-shape: 5n-42
The formula for the revered T-shape: 5n+42
T total of the reversed T-shape=5n+42
L.H.S.= 23+29+35+34+36
= 157
R.H.S.= 5n+42
=5(23)+42
=157
∵L.H.S.=R.H.S.
∴The reverse in the minus sign has worked.
The two examples above show that the formula for mirror image is 5n+7g.
Part 3 continued-Rotation (90° clockwise or anti-clockwise)
T- shape 1
T-shape2
T-shape 3
T number of T-shape1 and T-shape2: 21
T total of T-shape1 and T-shape2: 42/112 (difference=70)
T number of T-shape1 and T-shape2: 67
T total of T-shape1 and T-shape2: 272/342 (difference=70)
T number of T-shape1 and T-shape2: 67
T total of T-shape1 and T-shape3: 272/328 (difference=56)
T number of T-shape1 and T-shape2: 21
T total of T-shape1 and T-shape3: 42/98 (difference=56)
The formula for rotation of 90° clockwise: 5n-7g+70
The formula for rotation of 90° anticlockwise: 5n-7g+56
*These formulas only work for a 9x9 grid*
Part 3 continued-formula for rotation for all grid size
Take a 6x6 grid square as example:
The T number of both T shapes: 21
The T total of the T-shape2: 112
The difference between the 2 T-totals: 49
After my analysis on the two grids of different size, I found out that the formula for clockwise rotation is 5n-7g+7(g+1) while the formula for anti-clockwise rotation is 5n-7g+7(g-1).
Part 3 continued-Combination of transformation-Translation+Rotation
Vector(3,-1)
