Example:
T- number = T39 p = T-number
T- total = 258 q = T-total
Grid size = 9 x 9 r = grid size
Calculations:
p + p + p + p + p = 5p
r + 2r + 2r + 2r = 7r
-1 + 1 = 0
Formula for T-shape: q = 5p + 7r
Justification: (5 x 39) + (7 x 9) = 258
The examples above give evidence to justify that the formula (q = 5p + 7r) works for all T- shapes that are rotated 1800.
I will now alter the grid sizes to try and justify as to whether my formula works on different grid sizes.
T- number = T35 p = T-number
T- total = 231 q = T-total
Grid size = 8 x 8 r = grid size
Calculations:
p + p + p + p + p = 5p
r + 2r + 2r + 2r = 7r
-1 + 1 = 0
Formula for T-shape: q = 5p + 7r
Justification: (5 x 35) + (7 x 8) = 231
The example above justifies clearly that the formula (q = 5p + 7r) works for all T-shapes that are rotated 1800.
Now I am going to change the grid size to a completely contrasting size, this time the grid size is 5 x 5 and a contrasting T-number will be used.
T- number = T8 p = T-number
T- total = 75 q = T-total
Grid size = 5 x 5 r = grid size
Calculations:
p + p + p + p + p = 5p
r + 2r + 2r + 2r = 7r
-1 + 1 = 0
Formula for T-shape: q = 5p + 7r
Justification: (5 x 8) + (7 x 5) = 75
Although this time the T-number I have used is different from the numbers used before the outcome is still exactly the same. This justifies that the formula (q = 5p + 7r) works for any T-shape that has a rotation of 1800 regardless of the T-number or the grid size.
Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for T-shapes that are rotated 1800 is (q = 5p + 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.
9 x 9 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
8 x 8 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
5 x 5 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
My formula to calculate the T-total is the same for all of these grid sizes (when the T-shape is rotated 1800). This justifies my formula, (q = 5p + 7r).
T- number = T49 p = T-number
T- total = 182 q = T-total
Grid size = 9 x 9 r = grid size
Calculations:
p + p + p + p + p = 5p
-r + -2r + -2r + -2r = -7r
-1 + 1 = 0
Formula for T-shape: q = 5p – 7r
Justification: (5 x 49) - (7 x 9) = 182
Example:
T- number = T31 p = T-number
T- total = 187 q = T-total
Grid size = 9 x 9 r = grid size
Calculations:
p + p + p + p + p = 5p
-r + -2r + -2r + -2r = -7r
-1 + 1 = 0
Formula for T-shape: q = 5p – 7r
Justification: (5 x 50) - (7 x 9) = 187
The above examples justify that the formula (q = 5p – 7r) works for all standard T-shapes. Using this formula the T-total can be calculated with the need of only the T-number and grid size.
I am now going to use different grid sizes to try and justify my formula, (q = 5p – 7r).
T- number = T38 p = T-number
T- total = 134 q = T-total
Grid size = 8 x 8 r = grid size
Calculations:
p + p + p + p + p = 5p
-r + -2r + -2r + -2r = -7r
-1 + 1 = 0
Formula for T-shape: q = 5p – 7r
Justification: (5 x 38) - (7 x 8) = 134
The above example clearly justifies that the formula (q = 5p – 7r) works for all standard T-shapes. Below is another example but this time the grid size is 5 x 5.
T- number = T12 p = T-number
T- total = 25 q = T-total
Grid size = 5 x 5 r = grid size
Calculations:
p + p + p + p + p = 5p
-r + -2r + -2r + -2r = -7r
-1 + 1 = 0
Formula for T-shape: q = 5p – 7r
Justification: (5 x 12) - (7 x 5) = 25
Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for standard T-shapes is (q = 5p - 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.
9 x 9 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
8 x 8 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
5 x 5 general pattern:
As the T-numbers increase by 1, the T-total increases by 5.
My formula to calculate the T-total is the same for all of these grid sizes (standard T-shape). This justifies my formula, (q = 5p - 7r).
