So X = 5T-63
T = 20
X = 5x20-63
= 100-63
= 37
Now I will do two more to check to see if it will work anywhere on the grid.
T=43
X=5x43-63 =152
T=79
X= 5x79-63 = 332
60 + 61 + 62 + 70 +79 = 332
So this proves my formula is right as I have tested it for three different T-numbers on a 9x9 grid and have found that none of them have been incorrect.
So the full formula for a 9x9 grid is X=5T-63
Where X=The T-Total, T = The T-Number
When trying to fine out the T-Total in a 6 by 6-number grid, If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like the one shown below the 6x6 grid.
This time the centre column of the T-Shape is going up in 6’s because of the table size. The formula for this size grid is T-Total = T-13+T-12+T-11+T-6+T= 5T-42
Now to do as I did before I will test my formula out: -
With X = T-Totals and T= T-Number
T = 35
So X= 5x35 – 42
This means X should equal 133
The numbers in the T-Shape are 22,23,24,29,35 and if you add them all together this makes 133 which shows my formula is true, but just to make sure I will test it again.
T = 16
So X = 5x16 – 42
So X= 32
3 + 4 + 5 + 10 + 16 = 32
This shows that my formula is correct!
I have tested the formula for two different T-Numbers and the formula works on a 6x6 grid.
The full formula for this size grid is: - X=5T-42
Below is a grid with parts of the formula in. I will use this to figure out a formula for any T-Total on any width grid.
T-total = T-2G-1
T-2G
T-2G+1
T-G
T
X=5T-7G
If the Grid Width (= G), and the T-Number (= T), the T-Total (= X) is calculated by
X=5T-7G
Now I’m going to try the grid that I drew earlier which was G=9
The formula is X=5T-7G
Which changes to X=5T-7x9
X=5T-63, this shows that this rule has a relationship between itself and the rule for the grid.
I will now try the formula on a 7 by 7 grid.
X=5X17-49 (2+3+4+10+17=36) – This is what the numbers in the T-Shape added together equals.
X=85-49
X=36
I am now going to try this formula on a 3 by 5 grid.
X=5X8-21 (1+2+3+5+8=19) – This is what the numbers in T-Shape added together equals.
X=40-21
X=19
I am now going to try this formula on a 6 by 10 grid.
X=5X33-42 (20+21+22+27+33=123
X=165-42
X=123
Now I have worked the formula out for the T-Shape in the one position, I am going to translate the T-Shape to different positions and investigate the relationships between the T-Total, the T-number and the grid size.
As you can see below the T-Shape can be set out just like the previous one I drew but the formulas are flipped over.
T + (T+G) + (T+2G+1) + (T+2G) + (T+2G-1) = 5T + 7G
As you will see the rule is fairly similar but instead of a take-away sum it is an additional sum, I think this happens because usually when you flip something it is the opposite to what it was and when you flip the T-Shape it flips the rule.
I will now try this new formula on a 4 by 7 grid, using 10 as the T-Number.
7G+5T
X=7x4+5x10 (17+18+19+14+10=78)
=28+50
=78
I will now try it on a 8 by 3 grid.
7G+5T
x=7x9+5x2 (19+20+21+11+2=73)
=56+10
=73
I have tried this new formula for an inverted T and it works, the formula is: X=+5T
I will now find out a formula that has been translated 90 degrees anti-clockwise
T
T-1
T-2
T-2-G
T-2+G
5T-7 this is what all the squares added above equals and this should be the rule.
I will use this rule on a 5by5grid this should not affect any outcome of the end result, I will use 8 as my T-Number though
X= 5T-7=5x8-7
X= 40-7=33 (Numbers In T-Shape = 1+6+11+7+8= 33)
So my rule is correct!
I have tested this new formula for a side ways T the formula works it is: X=5T-7
Before I write the formula I will predict what it will be 5T+7
T
T+1
T+2
T+2+G
T+2-G
5t+7
The formula is 5T+7 My Prediction was correct!
I have tested this new formula for a side ways T the formula works it is: 5T+7
The formula for a this way up is: X=5T-7G
The formula for a this way up is: X=7G+5T
The formula for a this way up is: X=5T+7
The formula for this way up is: X=5T-7
