T-Total=80
T-Total=5x23-35
T-Total=115-35
T-Total=80
So now you know the formula for the 5 by 5 grid is T-Total=5T-35
Now I will find a formula to solve any T-Total on any size table.
N=Grid Width
Here is the full formula to work when solving out the formula for the T-Shape on a certain grid size.
T-Total=T+T-n+T-2n+T-2n-1+T-2n+1
T-Total=5T-7n
Now lets try that formula on the 5 by 5 grid.
T-Total=5T-7n
T-Total=5T-7x5
T-Total=5T-35
That is the formula to work on the 5 by 5 grid. I will now check to see if it works on a different size grid.
Here is a 4 by 6 grid
T-Total=5T-7n
7N=7xGrid width
T-Total=5T-7X4
T-Total=5T-28
To make sure this formula works I will find a T-Shape on the 4 by 6 grid and see if that formula works.
T-Total=47
T-Total=5T-28
T-Total=5x15-28
T-Total=75-28
T-Total=47
3.
Now I’ve completed most tasks I will now investigate the formula for different T-Shape positions. I will start of by investigating a 180° T-Shape.
Here is a 5 by 5 grid to test on.
The total amount of numbers inside the T-Shape is 3+8+12+13+14=50
The 3 at the top of the T-Shape will be called the T-Number.
Here is the T-Shape by itself. As you can see the numbers in the middle is going up in 5’s because the grid width is 5.
Here is the converted version so I can work out the formula for the T-Total.
T-Total=(T)+(T+5)+(T+10)+(T+9)+(T+11)
T-Total=5T+35
Now to check if the formula works:
T-Total=50
T-Total=5x3+35
T-Total=15+35
T-Total=50
Here is another check to make sure.
T-Total=105
T-Total=5x14+35
T-Total=70+35
T-Total=105
Now I will see another formula for another grid size, which is 4 by 6.
That will be the T-Shape I will be finding the T-Total formula.
Firstly here is the formula I need to use.
That is the converted version of the 4 by 6 grid size.
T-Total=(T)+(T+4)+(T+8)+(T+7)+(T+9)
T-Total=5T+28
Now to check if the formula works:
T-Total=63
T-Total=5x7+28
T-Total=35+28
T-Total=63
Here is another example for that formula:
T-Total=98
T-Total=5x14+28
T-Total=70+28
T-Total=98
Now I will figure out a formula to find the T-Total for an inverted T-Shape on any size grid.
N=Grid Width
Here is the full formula to work when solving out the formula for an inverted T-Shape on any size grid.
T-Total=(T)+(T+n)+(T+2n)+(T+2n+1)+(T+2n+1)
T-Total=5T+7N
Now lets try that formula on the 5 by 5 grid.
T-Total=5T+7N
T-Total=5T+7x5
T-Total=5T+35
As you can see if you go up to page 5-6 you will find that it works out as the same formula as the one on this page.
Now for a different sized grid I haven’t used before.
I will use a 3 by 7 grid size.
T-Total=5T+7N
7N=7xGrid width
T-Total=5T+7x3
T-Total=5T+21
To make sure this formula works I will find an inverted T-Shape on the 3 by 7 grid and see if that formula works.
T-Total=61
T-Total=5x8+21
T-Total=40+21
T-Total=61
Now that has been completed I will convert the T-Shape 90° to the right.
Here is a 5 by 5 grid to test on.
The Total amount of numbers inside the T shape is 11+12+13+8+18=62
The 11 at the left of the T-Shape will be called the T-Number.
Here is the T-Shape by itself. As you can see the numbers on the right are going down in 5’s due to the grid size, which is 5.
Here is the converted version so I can work out the formula for the T-Total.
T-Total=(T)+(T+1)+(T+2)+(T-3)+(T+7)
T-Total=5T+7
Now to check if this formula works:
T-Total=62
T-Total=5x11+7
T-Total=55+7
T-Total=62
Here is another T-Shape to make sure:
T-Total=97
T-Total=5x18+7
T-Total=90+7
T-Total=97
Here is another grid size to find out what formula I need to use for that specific grid size.
The grid size I’ll use will be 7 by 4.
Firstly here is the formula I’ll need.
T-Total=(T)+(T+1)+(T+2)+(T-5)+(T+9)
T-Total=5T+7
Now to check if the formula works:
T-Total=57
T-Total=5x10+7
T-Total=50+7
T-Total=57
Here is another T-Shape to make sure.
T-Total=102
T-Total=5x19+7
T-Total=95+7
T-Total=102
Now that has been completed I will obviously need to find the formula for a 90° to the right T-Shape on any size grid. Although I should solve the full working out, I found that the formula 5T+7 could be worked for all grid sizes. Here is an example on a 3 by 3 grid size, which you could use:
T-Total=27
T-Total=5x4+7
T-Total=20+7
T-Total=27
Now that has been completed I will be using a 270° to the right T-Shape on a 5 by 5 grid.
The 19 at the right of the T-Shape will be called the T-Number.
Here is the T-Shape by itself:
As you can see the numbers on the left is going down in 5’s because of the grid size.
T-Total=(T)+(T-1)+(T-2)+(T-7)+(T+3)
T-Total=5T-7
Now to see if the formula works:
T-Total=88
T-Total=5x19-7
T-Total=95-7
T-Total=88
Here is another check on the same grid:
T-Total=38
T-Total=5x9-7
T-Total=45-7
T-Total=38
Here I will change the grid size to find a formula. The grid size I’ll be using is 4 by 7.
This will be the T-Shape I will be using to find out if my formula works. I will firstly need to figure out a formula.
T-Total=(T)+(T-1)+(T-2)+(T-6)+(T+2)
T-Total=5T-7
Let me try this formula on my T-Shape:
T-Total=73
T-Total=5x16-7
T-Total=80-7
T-Total=73
Now let’s try this on another T-Shape in the same grid size:
T-Total=108
T-Total=5x23-7
T-Total=115-7
T-Total=108
Now this has been completed I will now try to find a formula to all T-Shapes rotated 270° to the right on any size grid. Although I should solve the full working out, I found that the formula 5T-7 could be worked for all grid sizes. Here is an example on a 9 by 8 grid:
T-Total of T-Shape=158
T-Total=5x33-7
T-Total=165-7
T-Total=158
Here is another example:
T-Total of T-Shape=123
T-Total=5x26-7
T-Total=130-7
T-Total=123
So now you understand that for the first T-Shape the formula to work out a formula to solve any T-Total on any size table is –
T-Total=5T-7n
Now you understand that for the inverted T-Shape the formula to work out a formula to solve any T-Total on any size table is –
T-Total=5T+7N
Now you understand that for the first sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –
T-Total=5T+7
Now you understand that for the second sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –
T-Total=5T-7