I will now work out the formula for the T-Shape on the 3 x 3 grid:
The formula worked out is 5n - 21.
The T-Number of the T-Shape is 8, therefore n = 8. I will now work out the T-Total:
T = 5n – 21
T = (5 x 8) – 21
T = 19
For 4 x 4 Grid Size:
This is the first of two T-Shapes that I will do for the 4 x 4 grid.
I will now work out a formula for all T-Shapes on a 4 x 4 grid:
The formula worked out is 5n - 28.
The T-Number of the T-Shape is 10, therefore n = 10. I will now work out the T-Total:
T = 5n – 28
T = (5 x 10) – 28
T = 22
I know that this is correct because:
T = 1 + 2 + 3 + 6 + 10
T = 22
This is the second grid I will now do from the 4 x 4 grid, to check that my formula is correct.
The T-Number of the T-Shape is 11, therefore n = 11.
I will now use my formula to work out that T-Total for this T-Shape:
T = 5n – 28
T = (5 x 11) – 28
T = 27
I know that this is correct because:
T = 2 + 3 + 4 + 7 + 11
T = 27
For 5 x 5 Grid Size:
This is the first of two T-Shapes that I will do for the 5 x 5 grid.
I will work out a formula for all T-Shapes on a 5 x 5 grid:
The formula worked out is 5n - 35.
The T-Number of the T-Shape is 12, therefore n = 12. I will now work out the T-Total:
T = 5n – 35
T = (5 x 12) – 35
T = 25
I know that this is correct because:
T = 1 + 2 + 3 + 7 + 12
T = 25
This is the second grid I will now do from the 5 x 5 grid, to check that my formula is correct.
The T-Number of the T-Shape is 13, therefore n = 13.
I will now use my formula to work out that T-Total for this T-Shape:
T = 5n – 35
T = (5 x 13) – 35
T = 30
I know that this is correct because:
T = 2 + 3 + 4 + 8 + 13
T = 30
For each grid, I have now worked out a formula that will find the T-Total of any T-Shape on that grid.
Now, I am going to incorporate the grid size into every formula.
For 3 x 3 grid, the formula for the T-Total is 5n - 21. By dividing 21 by the grid size, 3, and substituting it into the formula, I came up with the following:
T = 5n - 21
3
T = 5n – 7g
Where n is the T-Number and g is the grid size.
I will now test this formula on the 3 x 3 grid T-Shape.
The T-Number of the T-Shape is 8, therefore n = 8. I will now work out the T-Total using the new formula:
T = 5n – 7g
T = (5 x 8) – (7 x 3)
T = 40 – 21
T = 19
I already know that 19 is the T-Total for this T-Shape, so I know that my new formula is correct.
I then found that the same algebraic formula, 5n – 7g, could also be used for the 4 x 4 and 5 x 5 grids.
For the 4 x 4 grid, the formula for the T-Total is 5n – 28. I again put the grid size, 4, into the equation. This is what I did:
T = 5n - 28
4
T = 5n – 7g
Where n is the T-Number and g is the grid size.
I will now test this formula on the 4 x 4 grid T-Shape:
The T-Number of the T-Shape is 10, therefore n = 10. I will now work out the T-Total using the new formula:
T = 5n – 7g
T = (5 x 10) – (7 x 4)
T = 50 – 28
T = 22
I already know that 22 is the T-Total for this T-Shape, so I know that my new formula is correct.
I can now say that for any grid size, the formula 5n – 7g, where n is the T-Number and g is the grid size, will find the T-Total of any T-Number on any sized grid.
Now that I have found out a formula that incorporates g as the grid size, I will now try and add g into the T-shape. On the grid below, which for example is on a 4 x 4 grid, g = 4. Therefore every 4 in the formulas will be substituted for g:
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 transformations.
Now that I have investigated relationships between the T-total, T-numbers and grid size by translating the T-shape to different positions on the grid and changing the grid size. I will now use transformations in my investigation.
When using any grid size from 5x5 upward, the following is the formation of a T-shape when rotated at various angles, about the T-number.
I can now use this diagram to find expressions relating the T-total, T-number and grid size when rotating about the T-number at various angles.
Rotation 90° clockwise: (for any grid size)
I have now found that the expression for finding the T-total when the T-Shape is rotated 90º clockwise about point n is:
T = 5n + 7
However, I need to test whether my workings are correct. I will use the formula to work out the T-Total for this shape from a 3x3 grid:
The T-Number for this T-Shape is 4, therefore n = 4. I will now work out the T-Total using the new formula:
T = 5n + 7
T = (5 x 4) + 7
T = 20 + 7
T = 27
I know that this is correct because:
T = 3 + 4 + 5 + 6 + 9
T = 27
This will work on any grid size.
180º Rotation:
As I have already found in Part 2 that the expression relating to the T-total, T-numbers and grid size is:
T = 5n - 7g
I therefore predict that the expression for finding the T-total of a T-shape when it is rotated 180º will be
T = 5n + 7g
This is because the T-shape is in the opposite position, it is a reflection and the signs are reversed with the reversed position. Instead of moving upward from the T-number and having to subtract, I am moving downward which enables me to add instead. The numbers get smaller when moving upward, and larger when moving downward.
The T-Number for this shape from a 3x3 grid is 2, therefore n = 2. The grid size is 3, therefore g = 3. I will now substitute these into the inverse formula that I predicted would be correct for this rotation:
T = 5n + 7g
T = (5 x 2) + (7 x 3)
T = 10 + 21
T = 31
I know that this is correct because:
T = 2 + 5 + 7 + 8 + 9
T = 31
This proves that my prediction was correct and the expression works.
Rotation 90° Anti-Clockwise:
The expression is the opposite of the expression finding the T-total of a T-shape rotated 90º clockwise, which was 5n+7. This is anti-clockwise, therefore the formula is reversed and the sign changed.
Any Grid Size – Any T-Number – Anywhere on the Grid
Now that I have found the formulae for T-shapes when rotated either 90° or 180°, I will go on to try and find some formulae to work out the T-Total of any T-Shape, with any T-Number, on any size grid. I will do this using n as the T-Number, and g as the grid size.
Below is a T-shape from a 3x3 grid.
I will now substitute into this t-shape n and g, where n is the t-number and g is the grid size.
The T-Number of the T-Shape is 8, therefore n = 8. The grid size is 3, therefore g = 3. I will substitute these into the formulae to check if it is correct.
