How I got the formula
The formula starts with 5x the T-number. This is because The T-total rises by 5 for every T-number. For example look at the chart below. As the T-number rises by 1 the T-total rises by 5
Then I minus 63 which I got from by working out the difference between the T-number and another number in the T-shape. This has to be done wit all 4 numbers in the T-shape. After getting the difference of each number in the T-shape I then added them together and got 63.
Example
The T-number of the T-shape is 32. Now I will work out the difference between the T-number and the rest of the numbers in this T-shape.
32-13 = 19
32-14 = 18
32-15 = 17
32-23 = 9
TOTAL = 63
This will happen to all the shapes. To prove this I will do another.
The T-number is 66. Now I will work out the difference between the T-number and the rest of the numbers in this T-shape like I did on the last one.
66-47 = 19
66-48 = 18
66-49 = 17
66-57 = 9
TOTAL = 63
Once again the total turns out to be 63. This is where the 63 comes from in this equation. Another place 63 comes from is 9x7 = 63. The nine comes from the size of the grid. This grid is nine. For instance if the grid size was 13 by 13 then I would multiply 13x7. At the end of my coursework It will be visible to see that all the numbers we minus or plus will be divisible by seven. This is where I get the seven from. The seven works with all size grids. The other method that I found will also work with different size grids.
Now if I put all this together I get the formula
5T-number-63 = T-total
Using the formula
5x78-63 = T-total
5x78-63 = 327
Checking…
59+60+61+69+78 =327
As show the formula works because I have found the relationship between T-totals and T-numbers for the.
Part Two: I’ll be using grids of different sizes and then translate the T-shape in to different positions. Then ill investigate the relationship between the T-total, T-number and the grid size.
Here I am doing what I did in part one. This time ill be finding out more about the grid sizes and what they are capable of doing.
For this T-shape (11x11 Grid)
T-number = 24
T-total = 1+2+3+13+24 = 43
For this T-shape (9x9 Grid)
T-number = 20
T-total = 1+2+3+11+20 = 37
Even though T-shape looks to be in the same place the T-total and the T-number have risen. The T-number has risen by four and the T-total has risen by six. If I use the same rules I made in the part one I will get a new formula for the new size grid.
Using the long method
24-1 = 23
24-2 = 22
24-3 = 21
24-13 = 11
TOTAL = 77
Shorter method
7* 11 = 77
Multiply by eleven because that is the grid size
New formula
5T-number-77 = T-total
5x24-77 = 43
The same formula works with only changing the last number in the formula. Now I will try the same method on a smaller grid size to make sure that my method doesn’t only work when the grid size gets bigger.
T-number = 10
T-total = 1+2+3+6+10 = 22
7x4 = 28
Multiply by four because that is the grid size
5T-number-28 = T-total
5x10-28 = 22
By changing the size of the grids I‘ve learned that there is an overall formula for any size grid to find the T-total. This formula is 5T-number-7G-number. I found this formula out by combining the formula which I used to find the difference of the T-number with the formula that I’ve been using with the other grids. This formula is shown below.
5T-number-7G-number = T-total
5x32-7x7 = 111
To prove this formula works I’ll use the other method
32-17 = 15
32-18 = 14
32-19 = 13
32-25 = 7
5T-number-49 = T-total
5x32-49= 111
As show the formula work to make show that my formula works on all grids ill show it on a large grid.
5T-number-7G-number = T-total
5x125-7x13 = 534
To prove this formula works I’ll use the other method
125-98 = 27
125-99 = 26
125-100 = 25
125-112 = 13
5T-number-91=T-total
5x125-91=534
This shows that the method I have found works on a smaller scale and a bigger scale. I can know see that by changing the grid size I have to change the formula but still keep the rule of how I get the number to minus in the formula. Or I can use the overall formula which works for any size grid
Part Three: I’ll be changing the size of grids as well as doing transformations and combinations of transformations. I will also be investigating the relationship between the T-total, T-numbers, grid size and the transformations.
If I turn the T-shape around 180 degrees it will look like this. When I done this I realised if I reverse the T-shape I should have to reverse something in the formula.
It is obvious that I will have to change the minus sign to a different sign. I will try the opposite of minus which is plus.
5T-number + 63 = T-total
5x2 + 63 = 73
Checking…
T-number = 2
T-total = 2+11+19+20+21 =73
As show by turning the T-shape 180 degrees I had to change something in the formula. By changing the minus sign into a plus sign the formula worked.
Next I’ll move the shape on its side. I’ll keep the same formula that I had at the beginning. Once Again I’ll change the minus number. I can work out the number to minus by working out the difference in the T-number to each number in the T-shape.
12-1 =11
12-10= 2
12-19= -7
12-11 = 1
TOTAL = 7
Formula
5T-number – 7 = T-total
5x12 - 7= 53
Checking…
T-number = 12
T-total = 1 +10 +19 +11 +12 = 53
This formula has worked. Now if I rotate the t-shape 180 degrees, the same will happen, as what happened when the T-shape when it was turned 180 degrees from its first original position. To prove this it will be shown below.
5T-number + 7 = T-total
5x70 + 7 = 357
Checking…
T-number = 70
T-total = 70+71+72+63+81 = 357
Now that I have worked out all the formulas for the position in the normal sized T-shape. No I’m going to enlarging the T-shape. I will double the T-shape. The new shape is shown below on the 9x9 grid. I have added the numbers together in the squares of the T-shape. This leaves me with my original T-shape but with larger numbers in the grid.
176-24(1+2+10+11) = 152
176-32(3+4+12+13) = 144
176-40(5+6+14+15) = 136
176-104(21+22+30+31) = 72
TOTAL= 504
I have the rest of the formula. The formula is identical apart from the number we minus or plus.
Formula
5T-number –504 = T-total
5x176-504 = 376
I have proven that the formula works
Conclusion
In conclusion I have learned that in this project I have found out many ways in which to solve the problem I have with the T-shapes being in many different positions with different many sizes of grids. The way I have made the calculations of this project less difficult is was by creating a many main formulas that change for all the different circumstances.