T-number = 36 and T-total = 36 + 28 + 20 + 19 + 21 = 124
Y = 5T – 56
Y = 5(36) – 56
Y = 124
T-number = 55 and T-total = 55 + 47 + 39 + 38 + 40= 219
Y = 5T – 56
Y = 5(55) – 56
Y = 219
Y= 5T – 56
The formula for T-total is tested for 3 different samples, which is proved to be worked for an 8 by 8 grid.
Let G be the width of the grid.
After investigating the relationships between the T-number and the T-total of a 9 by 9 grid and an 8 by 8 grid, a T-shape like the one drawn below is found:
Therefore, a formula for any T-total on any grid of any width can be worked out from that:
Y = T + T – G + T – 2G + T – 2G – 1 + T – 2G + 1
Y = 5T – 7G
Test
To start with, the 9 by 9 grid drawn before is going to be tested.
Y = 5T – 7G
Y = 5T – 7(9)
Y = 5T – 63
This formula is exactly the same as the one, which is proved to be worked for a 9 by 9 grid.
A 6 by 6 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 22 and T-total = 22 + 16 + 10 + 9 + 11 = 68
Y = 5T – 7G
Y = 5(22) – 7(6)
Y = 68
A 3 by 3 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 8 and T-total = 8 + 5 + 2 + 1 + 3 = 19
Y = 5T – 7G
Y = 5(8) – 7(3)
Y = 19
Y= 5T – 7G
The formula for T-total is tested for 3 different samples, which is proved to be worked for any grid of any width.
Now the T-shape is going to be translated to different positions in grids of different sizes other than the original 9 by 9 grid.
Let us focus on the 8 by 8 grid drawn before.
Take this T- shape above as the original position.
If it is translated by the vector 2 , the new T-number and the T-total will be
-3
changed to 44 and 164 respectively.
If the T-shape is then translated by the vector 5 , the new T-number and the T-total will
-5
be changed to 63 and 259 respectively.
A formula of this translation of the T-shape can be worked out by working the vectors with the original T-number.
Let a be the vector and y be the new T-total after being translated.
b
a only moves horizontally which, in this case, only differs by 1 unit whether moving to the left or to the right.
b only moves vertically which, in this case, only differs by 8 units in this 8 by 8 grid whether moving up or down.
Therefore:
The formula for T-total in an 8 by 8 grid worked out before:
Y= 5T – 56
The formula for new translated T-shape in an 8 by 8 grid:
y = 5T – 56
y = 5(T + a – 8b) – 56
Test
Take this T-shape above as the original position.
If it is translated by the vector -3 , the new T-shape will be:
-2
The T-total of this new T-shape is 58 + 50 + 42 + 41 + 43 = 234
Let us see if the formula works.
y = 5(T + a – 8b) – 56
y = 5[45 + (–3)– 8(– 2)] – 56
y = 234
Take this T-shape above as the original position.
If it is translated by the vector 4 , the new T-shape will be:
4
The T-total of this new T-shape is 31 + 23 + 15 + 14 + 16 = 99
Let us see if the formula works.
y = 5(T + a – 8b) – 56
y = 5[59 + 4 – 8(4)] – 56
y = 99
After investigating the two formulae for new translated T-total for a 9 by 9 grid and an 8 by 8 grid, a general formula for new translated T-total for any grid of different sizes can be worked out:
Let G be the width of the grid.
y = 5(T + a – G b) – 7G
This general formula is basically based on the formula for T-total on any grid of any width worked out previously, and further developed by adding the vectors of the translation.
Part 3
Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the
T-total, the T-numbers, the grid size and the transformations.
(9 by 9 grid)
First of all, the T-shape is going to be translated in 90° clockwise. To start with, take the T-number as the centre of rotation.
The original position:
The new T-shape will be:
If the other numbers in the T-shape other than the T-number are taken away from the T-number, a T-shape like the one drawn below is found:
Unlike the position of the original T-shape, the centre column of this new transformed T-shape is increasing by 1 to the right from the T-number. Because of the grid size, the right column of the T-shape is decreasing by 9 up the column. Thus a formula can be worked out to find any T-total with the T-number given:
Y = T + T + 1 + T + 2 + T + 2 + G + T + 2 – G
Y = 5T + 7
Test
For the T-shape drawn before, the T-number is 20 and the T-total is:
20 + 21 + 22 + 13 + 31 = 107
Let us see if the formula works.
Y = 5T + 7
Y = 5(20) +7
Y = 107
Two more tests on this formula are to be done for accuracy.
A 6 by 6 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 27 and T-total = 27 + 28 + 29 + 23 + 35 = 142
Y = 5T + 7
Y = 5(27) + 7
Y = 142
An 8 by 8 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 41 and T-total = 41 + 42 + 43 + 35 + 51 = 212
Y = 5T + 7
Y = 5(41) + 7
Y = 212
Y=5T + 7
The formula for new transformed T-total in 90° clockwise with the T-number being the centre of rotation is tested for 3 different samples, which is proved to be worked for any grid of any width.
Then, the T-shape is going to be transformed in 180° clockwise taking the T-number as the centre of rotation.
The original position:
The new T-shape will be:
If the other numbers in the T-shape other than the T-number are taken away from the T-number, a T-shape like the one drawn below is found:
Because of the grid size, the centre column of the T-shape is increasing by 9 down the column from the T-number at the top. Thus a formula can be worked out to find any T-total with the T-number given:
Y = T + T + G + T + 2G + T + 2G + 1 + T + 2G – 1
Y = 5T + 7G
Test
For the T-shape drawn before, the T-number is 22 and the T-total is:
22 + 31 + 40 + 39 + 41 = 173
Let us see if the formula works.
Y = 5T + 7G
Y = 5(22) +7(9)
Y = 173
Two more tests on this formula are to be done for accuracy.
A 6 by 6 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 23 and T-total = 23 + 29 + 35 + 34 + 36 = 157
Y = 5T + 7G
Y = 5(23) + 7(6)
Y = 157
An 8 by 8 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 44 and T-total = 44 + 52 + 60 + 59 + 61 = 276
Y = 5T + 7G
Y = 5(44) + 7(8)
Y = 276
Y=5T + 7G
The formula for new transformed T-total in 180° clockwise with the T-number being the centre of rotation is tested for 3 different samples, which is proved to be worked for any grid of any width.
Then, the T-shape is going to be transformed in 90° anti-clockwise taking the T-number as the centre of rotation.
The original position:
The new T-shape will be:
If the other numbers in the T-shape other than the T-number are taken away from the T-number, a T-shape like the one drawn below is found:
The centre column of this new transformed T-shape is decreasing by 1 to the left from the T-number. Because of the grid size, the left column of the T-shape is decreasing by 9 up the column. Thus a formula can be worked out to find any T-total with the T-number given:
Y = T + T – 1 + T – 2 + T – 2 + G + T – 2 – G
Y = 5T – 7
Test
For the T-shape drawn before, the T-number is 34 and the T-total is:
34 + 33 + 32 + 23 + 41 = 163
Let us see if the formula works.
Y = 5T – 7
Y = 5(34) – 7
Y = 163
Two more tests on this formula are to be done for accuracy.
An 8 by 8 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 48 and T-total = 48 + 47 + 46 + 38 + 54 = 233
Y = 5T – 7
Y = 5(48) – 7
Y = 233
A 3 by 3 grid is going to be tested now.
Take the T-shape below as a sample:
T-number = 6 and T-total = 6 + 5 + 4 + 1 + 7 = 23
Y = 5T – 7
Y = 5(6) – 7
Y = 23
Y=5T – 7
The formula for new transformed T-total in 90° anti-clockwise with the T-number being the centre of rotation is tested for 3 different samples, which is proved to be worked for any grid of any width.
Now I am going to look at transformations with the centre box of the T-shape being the centre of rotation other than the T-number. We look at the 9 by 9 grid first.
The T-shape is going to be transformed in 90° clockwise.
Take the T-shape below as the original position:
If the centre of rotation is the box 11, the new T-shape will be:
As the working shown before, the formula for this new T-total will be:
Y = T + T + 1 + T + 2 + T + 2 + G + T + 2 – G
Y = 5T + 7
The only difference between this and the one that worked out before taking the T-number as the centre of rotation is that the T-number has changed from the original one, T, to
T – G + 1 which the general formula be:
Y = 5(T – G – 1) + 7
Similarly, for all other transformations, this new T-number, T – G + 1, can be substituted into the formulae for the transformations.
Therefore:
Transformation of 180° clockwise:
Y = 5T + 7G
Y = 5(T – 2G) + 7G
Transformation of 90° anti-clockwise:
Y = 5T – 7
Y = 5(T – G + 1) – 7
Now I have worked out the formulae for transformations of the T-shape in grids of any width. I am going to try other combinations of transformations of the T-shape.
Combining transformations with translations:
We focus on the combination of transformations with translation in a 9 by 9 grid.
First we look at the combination of translation with a transformation of 90° clockwise.
Take the T-shape below as the original position:
Take the T-number as the centre of rotation.
If the T-shape is first translated by the vector 4 , the new T-shape will be:
-2
The formula for this translation has been worked before:
y = 5(T + a – G b) – 7G
Then, if this translated T-shape is further transformed in 90° clockwise, the new T-shape will be:
The formula for this transformation has been worked before:
Y = 5T + 7
A general formula for this combination of translation with transformation can be worked out by putting these 2 formulae together and further developed. The formula for the new translated T-number, (T + a – G b), can be substituted into the T-number of the formula for the transformation, Y = 5T + 7.
Then, the new general formula is:
Y = 5(T + a – G b) + 7
Test
Take the T-shape below as the original position:
If this T-shape is translated by the vector 5 , and then followed by a transformation of
-1
90° clockwise, the new T-shape will be:
T-number = 34 and T-total = 34 + 35 + 36 + 27 + 45 = 177
Let us see if the formula works.
Y = 5(T + a – G b) + 7
Y = 5[20 + 5 – 9 (–1)] + 7
Y = 177
Then, we now look at the combination of translation with a transformation of 180° clockwise.
Take the T-shape below as the original position:
Take the T-number as the centre of rotation.
If the T-shape is first translated by the vector 2 , the new T-shape will be:
-1
The formula for this translation has been worked before:
y = 5(T + a – G b) – 7G
Then, if this translated T-shape is further transformed in 180° clockwise, the new T-shape will be:
The formula for this transformation has been worked before:
Y = 5T + 7G
Similar to the previous combination, a new general formula can be worked out from those 2 formulae of translation and transformation:
Y = 5(T + a – G b) + 7G
Test
Take the T-shape below as the original position:
If this T-shape is translated by the vector 6 , and then followed by a transformation of
-1
180° clockwise, the new T-shape will be:
T-number = 35 and T-total = 35 + 44 + 53 + 52 + 54 = 238
Let us see if the formula works.
Y = 5(T + a – G b) + 7G
Y = 5[20 + 6 – 9 (–1)] + 7(9)
Y = 238
Then, we now look at the combination of translation with a transformation of 90° anti-clockwise.
Take the T-shape below as the original position:
Take the T-number as the centre of rotation.
If the T-shape is first translated by the vector 3 , the new T-shape will be:
-2
The formula for this translation has been worked before:
y = 5(T + a – G b) – 7G
Then, if this translated T-shape is further transformed in 90° anti-clockwise, the new T-shape will be:
The formula for this transformation has been worked before:
Y = 5T – 7
Similar to the previous combinations, a new general formula can be worked out from those 2 formulae of translation and transformation:
Y = 5(T + a – G b) – 7
Test
Take the T-shape below as the original position:
If this T-shape is translated by the vector 7 , and then followed by a transformation of
-5
90° anti-clockwise, the new T-shape will be:
T-number = 72 and T-total = 72 + 71 + 70 + 61 + 79 = 353
Let us see if the formula works.
Y = 5(T + a – G b) – 7
Y = 5[20 + 7 – 9 (–5)] – 7
Y = 353
Now I am going to look at combinations of transformations with others being the centre of rotation other than the T-number.
(9 by 9 grid)
Basically, the combinations can be regarded as separate translations and transformations. Therefore, it is the same as the previous working of the transformations with other places as the centre of rotation other than the T-number.
Transformation of 90° clockwise:
Y = 5T + 7
Y = 5(T – G + 1) + 7
Transformation of 180° clockwise:
Y = 5T + 7G
Y = 5(T – 2G) + 7G
Transformation of 90° anti-clockwise:
Y = 5T – 7
Y = 5(T – G + 1) – 7
Then, the general formulae for all combinations of transformations and translations with the centre box of the T-shape as the centre of rotation other than the T-number will be:
Combination of transformation of 90° clockwise:
Y = 5(T + a – G b) + 7
Y = 5(T + a – G b – G – 1) + 7
Combination of transformation of 180° clockwise:
Y = 5(T + a – G b) + 7G
Y = 5(T + a – G b – 2G) + 7G
Combination of transformation of 90° anti-clockwise:
Y = 5(T + a – G b) – 7
Y = 5(T + a – G b – G + 1) – 7
These formulae are formed by merging the 2 parts, T + a – G b and T – G + 1 of the formulae for translation and transformation with the centre box of the T-shape being the centre of rotation respectively.
Test
Using a 7 by 7 grid this time.
Take the T-shape below as the original position:
Take the centre box as the centre of rotation.
If this T-shape is translated by the vector -2 , and then followed by a transformation of
0
90° clockwise, the new T-shape will be:
T-number = 8 and T-total = 8 + 9 + 10 + 3 + 17 = 47
Let us see if the formula works.
Y = 5(T + a – G b – G – 1) + 7
Y = 5(18 + (–2) – 7(0) – 7 – 1) + 7
Y = 47
Using the same original position, if it is translated by the vector 2 , and then followed by
-1
a transformation of 180° clockwise, the new T-shape will be:
T-number = 13 and T-total = 13 + 20 + 27 + 26 + 28 = 114
Let us see if the formula works.
Y = 5(T + a – G b – 2G) + 7G
Y = 5[18 + 2 – 7(–1) – 2(7)] + 7(7)
Y = 114
If it is then translated by the vector 0 , and then followed by a transformation of 90°
-3
anti-clockwise, the new T-shape will be:
T-number = 33 and T-total = 33 + 32 + 31 + 24 + 38 = 158
Let us see if the formula works.
Y = 5(T + a – G b – G + 1) – 7
Y = 5(18 + (0) – 7(–3) – 7 + 1) – 7
Y = 158
These formulae are tested for 3 different samples and are proved to be worked combinations of transformations of those 3 angles.
Conclusion:
Final mark: 8 8 8 (full mark)
