N = 20
T = (5x20) – 63
= 100 – 63
= 37
The above result corresponds with that of the result attained earlier. Below, I have attempted the same thing with a different T number N = 53
T = (5x53) – 63
= 265 – 63
= 202
Yet again, the equation has produced a correct answer. I have tried again below to achieve the same affect.
N = 24
T = (5x24) – 63
= 120 –63
= 57
After a few more attempts, I came to the conclusion that the equation “T = 5N – 63” is correct for the 9 by 9 grid.
8 by 8 Grid
As I did with the 9 by 9 grid, I worked out the t-total for various t-shapes in the grid the long way (adding each individual digit in the t-shape, one by one). I will the see if the equation for the 9 by 9 grid would work on this grid.
1 + 2 + 3 + 10 + 18 = 34
5 + 6 + 7 + 14 + 22 = 54
33 + 34 + 35 + 42 + 50 = 194
37 + 38 + 39 + 46 + 54 = 214
Then I attempted to use the equation in the previous grid to work out the t-total of a t-shape.
T = 5N - 63
N = 22
T = (5x22) – 63
= 110 –63
= 47
As we can see above, the result was wrong. So I decided to do what I did in the previous grid and work out a new equation.
Now I bring all the terms together to get an equation.
T = N – 17+N – 16+N – 15+N – 7+N
= 5N – 56
I then check if the equation is correct.
N = 18
T = (5x18) – 56
= 34
We find that this new equation work in this case, but try it with a different t-number to confirm its validity.
N = 50
T = (5x50) –56
= 194
Again, the equation has produced another correct result. I try it again one more time to make sure.
N = 22
T = (5x22) – 56
= 54
So I have produced another equation that works with in this grid.
7 by 7 Grid
Below are some totals for t-shapes on the 7 by 7 grid on the right.
1 + 2 + 3 + 9 + 16 = 31
5 + 6 + 7 + 13 + 20 = 51
29 + 30 + 31 + 37 + 44 = 171
33 + 34 + 35 + 41 + 48 = 191
Now I will test the two previous equations to find out if they work on this grid.
T = 5N – 63
N = 16
T = (5x16) – 63
= 80 – 63
= 17
The above equation (which was for the 9 by 9 grid) proved to be incorrect for this grid. Below is the equation for the 8 by 8 grid.
T = 5N – 56
N = 16
T = (5x16) – 56
= 80 – 56
= 24
Again, the equation doesn’t work on this grid. Like I did in the two previous grids, I will work out a new equation.
The equation for the t-shape above is: -
T = N – 15 + N – 14 + N – 13 + N – 7 + N
= 5N - 49
I must test the equation as I did on the previous grids.
N = 16
T = (5x16) –49
= 80 – 49
= 31
The equation has produced its first correct answer. I will carry on and test the next t-shape I know the t-total for.
N = 20
T = (5x20) – 49
= 100 – 49
= 51
As expected, the equation has produced yet another correct answer. Another example is below.
N = 44
T = (5x44) – 49
= 220 – 49
= 171
Here is the last equation I will show from the 7 by 7 grid to show that the equation is correct.
N = 48
T = (5x48) – 49
= 240 - 49
= 191
6 by 6 Grid
To start this grid off, I will work out 4 t-totals.
1 + 2 + 3 + 8 + 14 = 28
4 + 5 + 6 + 11 + 17 = 43
19 + 20 + 21 + 26 + 32 = 118
22 + 23 + 24 + 29 + 35 = 133
Below, I have illustrated how I found out the equation for this grid.
The equation for the above is as follows: -
T = N – 13 + N – 12 + N –11 + N – 6 +N
= 5N – 42
Just to make sure the equation is correct, I will test it.
N = 14
T = 5N – 42
= 70 – 42
= 28
Below is another example of how the equation is correct.
N = 35
T = 5N – 42
= 175 – 42
= 133
So now I have determined that the equation is correct.
Analysis
After investigating, I have managed to come up with the table below.
What I have seen through this table is that all the equations from each grid all contain “5N” minus something. I don’t believe this is so for no reason. Also, if you look closely, the last number of each equation, they are of the 7 times table e.g. 63 = 7 x 9 or 42 = 7 x 6. What we now notice from this is that if you divide the last number in each equation by 7, you will see that the result would be the grid size. An example of this is shown below.
For the 9 by 9 grid the formula is: -
T = 5N – 63
63 / 7 = 9
So now what I can say is to figure out the t-total of a t-shape in any size grid, you need the formula “T = 5N – X”. The way you find out the t-total is as follows.
You firstly choose the t-shape you wish to find the t-total for. You then look at the grid size e.g. 7 by 7, and take the “7” from the grid size and multiply it by 7. This is your “X” value. You then multiply the t-number of the t-shape you are working out the t-total for and subtract the X value away from it. You should see, if done correctly, that the result would correspond with that of the real t-total.
What you can also figure out with this method is the t-total of an upside-down t-shape. To do this, you do as above, but instead of subtracting the X value away from 5N, you add it on. Note that all of the above equations have been based on upright t-shapes and don’t work if the t-shape is on its side.
