The T-number is 76.
I’m going to use the formula to find the T-total.
(Formula 1): Tt = 5n - 63
n = 76
Tt = (5 * 76) – 63
= 380 - 63
= 317
Tt = 76 + 67 + 57 + 58 + 59
= 317 correct!
This shows that the formula works as the T-total was acquired correctly by using the formula.
In conclusion, this formula works out any Tt of a T-shape in this 9 by 9 grid. This formula is made up of 5n – 63. 5 is the amount of numbers in the T-shape, this is multiplied by n. This total is then subtracted by 63. 63 is taken from the addition of the numbers that n is subtracted from in the T-shape where n is used. I.e. 19 + 18 + 17 + 9 = 63
The formula for the T-total is shown here:
Tt = 5n - 63 (formula 1)
2. use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total, T-number and the grid size.
In the previous question, if we look at the differences between the numbers in the T. We find that the first difference is the same as the width of the grid.
The first difference is the width of the grid because as we go up in the T, the next number will be the first number – the width. Consequently, we can substitute the first difference (9) with x (being the width).
In the T shown here, I have substituted the 9 with x. this shows the width of the grid.
To find the T-total of this T, I have to add all the formulas together.
Tt = n-(2x-1) + n-2x + n-(2x+1) + n-x + n
= 5n – 7x (formula 2)
This formula is made up of two constants which are the 5 and 7. The 5 is taken from addition of the number of n’s used in this T. The 7 is taken from the addition of the x’s which are subtracted from n. I think that the first subtraction will be the width of the grid in any T-shape, therefore I have used x in this formula to show the width of the grid.
Now I must test to see if this formula works. I am going to use the first T in question 1, this is drawn on a 9 by 9 grid.
This T-shape has a T-number of 20.
The width of the grid is 9.
Substitute these into formula 2 to find the T-total.
Tt = 5n – 7x
= (5 * 20) – (7 * 9)
= 100 - 63
= 37
Tt = 1 + 2 + 3 + 11 + 20
= 37 It works!
To show this formula works, I am going to use a T-shape in a 10 by 10 grid.
Look at this T-shape. The T-number is 56. The width of the grid is 10.
Substitute these numbers into formula 2.
Tt = 5n – 7x
= (5 * 56) – (7 * 10)
= 280 - 70
= 210
Tt = 35 + 36 + 37 + 46 + 56
= 210 It works!
To show this formula works, I am going to use a T-shape in a 11 by 11 grid.
Look at this T-shape. The T-number is 63. The width of the grid is 11.
Substitute these numbers into formula 2.
Tt = 5n – 7x
= (5 * 63) - (7 * 11)
= 315 - 77
= 238
Tt = 63+ 52 + 40 + 41 + 42
= 238 It works!
This shows us that formula 2 works in finding out the T-total of a T-shape when one knows the width of the grid and the T-number of the T-shape.
Formula 2:
Tt = 5n - 7x
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.
I am now going to try and find relationships with grid sizes and the 180 degrees rotation of the T-shape with other grid sizes and then compare them.
Starting with a Grid 5 by 5.
n = 12
Red Tt = 12 + 7 + 1 + 2 + 3
= 25
180 degree clockwise rotation:
Black Tt = 12 + 17 + 21 + 22 + 23
= 95
Difference between the two Tt is: 70
Now I’m going to see what happens on a 6 by 6 width Grid
n = 14
Red Tt = 14 + 8 + 1 + 2 + 3
= 28
180 degree clockwise rotation:
Black Tt = 14 + 20 + 25 + 26 + 27
= 112
Difference of 84
Now I’m going to see what happens on a 7 by 7 Grid
n = 16
Red Tt = 16 + 9 + 1 + 2 + 3
= 31
180 degree clockwise rotation:
Black Tt = 16 + 23 + 29 + 30 + 31
= 129
Difference of 130
I am now going to put the Tt’s in a table to find relationships between the differences between the two Tt’s.
I have noticed that the difference between the two T-total’s goes up by 14 as the grid total goes up by 1. From this, I worked out that the grid width multiplied by 14 is equal to the difference.
5 * 14 = 70
6 * 14 = 84
7 * 14 = 98
Therefore, the formula for the difference between two T-totals is the grid number multiplied by 14. 14 is the constant here and the variable is the width (x)
Difference between two Tt’s = 14x (formula 4)
We know how to find the first Tt by using formula 2. Now we can work out the second Tt by adding formula 4 to the formula 2:
Second T-total = formula 2 + formula 4
= (5n – 7x) + 14x
= 5n + 7x (formula 5)
Consequently, to work out another T-shapes 180 degree transformation, one must know the T-number and the T-total of the shape in its first position.
e.g. using a 9 by 10 grid and a T-number of 33
Red Tt = 5n – 7x (formula 2)
= (5 * 33) – (7 * 9)
= 165 –
= 102
If we rotate this T-shape 180 degrees clockwise from the same T-number, we can work out the difference between the two T-totals.
Difference between T-total = 14 * 9
= 126
Black Tt = 102 + 126
= 228
Is this correct?
Black Tt = 33 + 42 + 50 + 51 + 52
= 228 it works.
I am now going to see if formula 5 works:
Black Tt = 5n + 7x
Substitute n = 33
x = 9
Black Tt = (5 * 33) + (7 * 9)
= 165 + 63
= 228 It works!
Show that formula 5 works in another grid.
n = 28 Red T
x = 11
After 180 degree clockwise rotation
Black Tt = 5n + 7x
= 5 * 28 + 7 * 11
= 217
Black T
check:
Black Tt = 28 + 39 + 49 + 50 + 51
= 217 It works!
I have shown that formula 5 works to find out the Tt of a rotated T-shape. Therefore, we can use formula 5 to work out the Tt of any T-shape which has been rotated by 180 degrees by its T-number. We must know the T-number and the Width of the grid to work this out.
I am now going to investigate relationships between T-shapes that are reflected through a line of symmetry at the left hand side of the first T.
The thick line is the line of symmetry.
The black T has a Tt of:
Black Tt = 1 + 2 + 3 + 9 + 16
= 31
I am going to compare this Tt to the Red Tt reflected in the line of symmetry.
Red Tt = 4 + 5 + 6 + 12 + 19
= 46
Red Tt - Black Tt = 46 – 31
= 15
Now I am going to investigate in another Grid and another reflection. I am going to use a 9 by 9 grid.
Black Tt = 11 + 12 + 13 + 21 + 30
= 87
Red Tt = 14 + 15 + 16 + 24 + 33
= 102
Red Tt – Black Tt = 102 – 87
= 15
I have used two different grids and found that the line of symmetry leads to a difference of 15 between the two T-totals.
Consequently I can derive a formula from this. I can use formula 2 to find out the Tt of the first T-shape then add 15 to this to get the second Tt after reflection. This works only using this line of symmetry. The 15 in this formula is derived because as I have shown, the numbers in the T-shape go up 15 from T-shape to the reflected T-shape:
Formula 6:
Tt of reflected T = Formula 2 + 15
= 5n – 7x + 15
Now I have to show that formula 6 works on another grid.
Black Tt = 26 + 27 + 28 + 38 + 49
= 168
Reflected Tt = 5n – 7x + 15
= (5*49) – (7*11) + 15
= 183
Check
Red Tt = 29 + 30 + 31 + 41 + 52
= 183 It works!
In conclusion I have found that formula 6 can be used to find the Tt of a reflected T-shape when we know the width of the grid and the T-number of this T-shape.
Furthermore, I think that formula 2 is a very important formula. Tt = 5n + 7x. This shows us that any T-shape which is upright can have its T-total worked out when we know the T-number and the width of the grid it is in. Then to work out transformations of this T-shape, other formula’s can be produced by adding or subtracting something from this formula. E.g. formula 6, where 15 is added to this formula to signify that the Tt goes up 15 in the reflection of the T-shape.
I have shown that the T-total of a T-shape is dependant on the width of the grid and it’s T-number.
