From looking at the centre column of a T-shape you can see that the numbers go up in 9’s because of the size of the grid therefore with a table like this a formula can be made to work out any T-total on this grid.
Below is the working for this formula:
T-total = T-19+T-18+T-17+T-9+T
= 5T-63
I am now to test this formula to see if it works.
For T-total I will use the letter X.
For the T-Number I will use the letter T.
T-Number=20
X = 5T-63
T = 20
X = 5x20-63
= 100-63
= 37
T-Number=21
X = 5T-58
T=21
X = 5x21-58
= 100-58
= 42
T-Number=22
X = 5T-53
T=22
X = 5x22-53
= 100-53
= 47
The above calculations show that the formula is correct.
I am now going to test this formula on a different sized grid. The size grid I have chosen to do next is an 11x11 grid.
T-total = 1+2+3+13+24 = 43
T-number = 24
The T-Total and the T-Number have risen even though the T-Shape looks to be in the same place. The T-Number has risen by four and the T-Total has risen by six. I am now going to test to see if the formula used in the previous grid works.
The above grid has been done the same was as I did in the 9x9 grid. I am now going to show the complete version of calculations for the formula.
24-1= 23
24-2 = 22
24-3 =21
24-13 =11
TOTAL =77
OR
7 x 11 = 77 (11=Grid Size)
The New formula for this Grid should be 5tn – 77= T-Total
5 x 24 - 77= 43
The same formula works with only changing the last number in the formula. I will now try the formula on a smaller grid size to make sure it is not because the grid size is bigger than the 9x9 grid.
T-number = 10
T-Total = 1+2+3+6+10= 22
7 x 4 = 28 (4=Grid Size)
5tn - 28 = T-Total
5 x 10 – 28 = 22
The formula has been proven to work on a smaller grid. I can see this by changing the grid size we have had to change the formula by changing the number that represents the grid size, but still shows the formula works.
I can find a formula to allow me to work out any T-total on the grid.
As the T-Total increases by 5, the beginning of the formula will be 5T
The formula for this grid will be:
T-Total= 5T – 63
I will now use my formula to predict the T-Total for T-Number 69.
T-total = 5 x 69 – 63
= 345 – 63
= 282
I will now check my calculations by drawing the appropriate T-shape.
T-Total= 50+51+52+60+69=282
My formula has been proved to work and my calculations correct.
By looking at my previous results, I have now decided that I am going to try and generalise for a G x G grid.
T-Total = T-2g-1 + T-2g + T-2g+1 + T-G + T
= 5T-7G
I will now use my general formula to predict the T-Total of T-Number 24 on a 7 x 7 grid.
T= 24 G=7
T-Total = 5 x 24 – 7 x 7
= 120 – 49
= 71
I will check this by drawing the appropriate T-shape and adding up the numbers within.
I am now going to increase the size of my T-shape and also choose a different size grid for this larger T-Shape.
The grid size I have chosen to do is a 10 x 10 grid. I am going to enlarge my T-shape to see if my formula will still work on a larger T-shape.
