I then worked out the T-Total for both shapes.
Green T-Shape: 1+2+3+11+20 = 37
Pink T-Shape: 2+3+4+12+21 = 42
Then, I calculated the T-Totals for 5 adjecent T-Shapes on my 9x9 grid and put them in tabular form.
I then noticed a pattern.
It appeared that everytime the T-Number went up by one, the T-Total went up by five (and vise versa if it was to go down and left). I then made a prediction
The Prediction (in red) : That if my thoery was correct for the T-Number 24 the
T-Total would be 57.
I then calculated this and my prediction turned out to be correct.
This is because there is 5 numbers in the T-Shape and each number in the T-Shape has to go up by one when it it translated to the right therefore adding five more onto the total. For this same reason, five has to be included as the main focus of my algebraic formula.
The Algebraic Formula
I could then, using the process of elimination and trial and error come up with an algebraic formula. However, I concluded that there must be a more logical way.
I therefore looked at my grid again.
Starting from my T-number, which in this case is called n, I have calculated this algrbraic t-shape. in the boxes of the shape, the numbers inside are related to the T-Number (n).
In this case on my 9x9 grid, the square directly above the T-Number (which is 20) is nine less. This makes that square n-9. The same method is used for the rest of the numbers in the shape.
To find the T-Total in a normal grid you have to add all of the numbers
In the T-shape together. Therefore I followed the same principal with
my algebraic T-shape.
This was my sum:
(n-19) + (n-18) + (n-17) + (n-9) + n = 5n – 63.
This also incorporates my earlier idea that 5 had to be the main focus of my formula.
To prove this I have developed this formula which is basically the reverse of what I have just done:
(5 x 20) – 63 = 37
Just to make sure, I am going to try my formula on one more T-Shape on the 9x9 grid before I proceed with my investigation to the 8 x 8 sized grid.
T-Number = 21
T-Total = 42
My Formula:
(5 x 21) – 63 = 42.
Now I know that it works for the 9 by 9 grid indefinitely, I will investigate other grid sizes to see if the formula works.
The 8 x 8 Grid.
I am now going to try to prove that my formula works on a 8 x 8 grid.
I have chosen two T-Totals to try my formula on.
Firstly I had to work out the T-Total for both of my shapes (shown in tabular form right)
I then used my formula to see if it was correct, I began with my “pink” T-Total.
My current formula = 5n – 63
To tell the vadility of my formula I tried it out:
(5 x 18) - 63 = 27
As I have just shown, the formula does not work with the 8 x 8 grid. So, I have to find a universal formula that can include all of the grid sizes.
To get this I havedecided to go back to my 9 x 9 grid.
I know that I must incorporate the grid number into the formula (in this case 9). I will divide 63 by the number in the top right corner, i.e. the grid number (g), 9. That leaves me with the formula:
5n – 7t = T-Total.
I will now go back to the T-Totals from my 9 x 9 grid and try my new and improved formula on both of them.
9 x 9 T-Shape One.
T-Number = 20
T-Total = 37
For this Formula I am going to explain why this works.
(5 x 20) – (7 x 9) = 37.
9 x 9 T-Shape Two.
T-Number = 21
T-Total = 42
(5 x 21) – (7 x 9) = 42.
So far, with the 9 x 9 grid, the formula works. I am now going to try it out on the 8 x 8 grid T-Totals.
8 x 8 T-Shape One.
T-Number = 18
T-Total = 34
(5 x 18) – (7 x 8) = 34
8 x 8 T-Shape Two.
T-Number = 19
T-Total = 39
(5 x 19) – (7 x 8) = 39.
As the formula has worked on both grids.
This has indefinatly proved again that my formula is correct.
To prove this fianally I am going to test my formula on two T-Shapes from a 7x7 grid
I have chosen two T-Shapes to try my formula on.
7 x 7 T-Shape One.
T-Number = 16
T-Total = 31.
(5 x 16) – (7 x 7) = 31
7 x 7 T-Shape Two
T-Number = 17
T-Total = 36
(5 x 17) – (7x7) = 36.
I now have a formula that relates the T-Number to the T-Shape regardless of the T-Shapes size as long as the formula is followed corectly.
