To find the T-total for this general T, you must add all the ‘n’s together. If we give the T-total the symbol ‘t’, the formula is:
t = 5n – 63
This can be rearranged for if you have the T-total and want to work out the T-number from it. That formula would be:
n = (t + 63)/5
This formula solves the first part of the task, however there are some restraints that need to be taken into account. The number ‘n’ cannot be absolutely anywhere in the table, as the rest of the T shape would not be able to fit. Therefore, this equation is only valid if the T-number can be found within the second column to the eighth column and within the third row to the ninth row. Any further out and some of the cells of the T would not fit into the grid and the formula would not work.
Part 2
Use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-totals, the T-numbers and the grid size.
To start with, it is obvious that a new general formula will be required to work with grids of different sizes as the last one relied quite heavily on the fact that the grid was 9 by 9. To do this all we have to do is substitute. If we call the new width of the grid ‘w’, then all that is needed is to replace the nine in the formula grid with ‘w’, so that it looks like this:
By adding these all together, we can get a new general formula for finding out t:
t = 5n – 7w
There is also a formula for finding out the T-number:
n = (t + 7w)/5
And finally, a new equation for working out the width of the grid when you are given a non-specific T shape:
w = (5n – t)/7
Again, there will be constraints but this time we will need to make them relevant for the size of the grid. The number ‘n’ will need to be within the 2nd and the ‘w-1’th column, and between the 3rd and ‘w’th row. This again will ensure that the T-shape fits entirely within the grid and the formula will work.
Also, this task calls for translations to be made around the grid, and note the difference that is made on the T-totals. After doing a few examples of this with real numbers, I decided that the best way to get real proof would be to try the task algebraically. Starting with my general formula I decided I would work with the vector (x,y). This would mean that you could move the T-shape any number across and any number down (within the constraints of the grid) including minus numbers and the formula would still work. Firstly I drew out a diagram which would help me see what I was doing:
To
The letters ‘+x + wy’ is added to the end because the x is how much it moves across and ‘wy’ demonstrates the amount it has moved up multiplied by the number in each row so that you get the additional number which is taken away.
The t-total will go from:
t = 5n – 7w
to
t = 5n – 7w + 5x - 5wy
To work out if this formula worked; I decided to test it out on a random T-number and vector to discover whether this was the case. The original T-number I decided on was 43 on a 10 by 10 grid. This makes the T-total 145. According to my formula, if I translate it by the vector (3,1), then the new T-total should be 110. If I actually do the translation, the new T-number would be 36 which, using my formula and working it out from the grid, has the T-total of 110, meaning that it is very likely that the formula works.
Part 3
Firstly, I made a list of possible transformations which could be done to the T-shape. These were:
- Rotations
- Reflections
- Enlargements
- Translations (covered in part 2)
- Combinations
After this, I began to think how the formula would change if the T was rotated. In a 90° rotation clockwise, the base of the T and the T-number would be to the left. This would change the general formula as the next number up in the T would be one more than the T-number, as would the one next to it. The numbers on the branches of the tree would equal each other out, one adding the grid size and on subtracting it. This would mean that the only thing that the grid size would have an effect on would be the constraints. A general T would look like this:
Overall by adding them together you would end up with the general formula
t = 5n + 7
To find out the formula for the T-number you rearrange the formula to get
n = (t – 7)/ 5
In this case the restraints would also be different. The n would have to be in or between the columns ‘1’ to ‘w-3’ and in the rows ‘2’ to ‘w-1’.
A similar thing would happen were the T rotated anticlockwise through the same angle, although much of the formula would be reversed. The T would now look like this:
The general formula would now be
t = 5n – 7
and the formula to find the T-number would be
n = (t + 7)/ 5
Having worked out this, I also realised that the formula for the translations would have changed with these rotations. The combination of rotation and translation might be a useful one to work out, so I started off with what the clockwise 90° rotation and did the same thing as I had done for translations before because it had worked the first time.
to
From doing this it is obvious that the same step of adding ‘+x-wy’ to the end of each equation, and therefore the general formula will add ‘+5x-5wy’ to it as it did before. In this way the general equation would be:
t = 5n + 7 + 5x – 5wy
The same addition of 5x – 5wy would also work on the formula for the other rotations of the shape as you are doing the same basic transformation and translation and therefore it will not make a difference.
Another transformation which I thought that I could investigate the relationship of was reflections. If you reflect the shape within the T-number, you get an upside-down T-shape which looks something like this:
I have already filled it in, based on the variables and properties discussed earlier. The numbers can be added up in this to also give basic formulas of
t = 5n + 7w
and
n = (t – 7w)/ 5
Through some more examples I decided that the formula for translations also worked on this as well, making the formula for reflected, translated Ts
t = 5n + 7w + 5x -5wy
The final transformation which I wanted to investigate into was enlargements. As before, I decided that it would probably be best to try to work it out using a general algebraic formula rather than working through lots of examples as there is unlikely to be a direct proportion between the T-number and the T-total. In my example I decided to enlarge the T to double the size by multiplying the width of the T by 2. I would keep the same T-number.
This T has the general formulas of
t = 20n – 66w + 10
n = (t + 66w – 10)/ 20
w = (20n – t + 10)/ 66
I also saw that with this T-shape, the formula for translations would also change because there are so many more squares in the T. Instead of adding ‘+5x-5wy’ you would now add ‘+20x-20wy’. This is because there are 20 squares in the T-shape rather than the 5 that there were before.
Conclusion
Overall, there is a relationship between the T-number and the T-shape with whatever alterations I have made to it and general formulas can be made using up to five variables. The most important general formulas which I have found have been highlighted in yellow as I feel that these are the most important results from my investigation.
