The total of the numbers inside the T-shape is 1+2+3+11+20=37
This is called the T-total.
The number at the bottom of the T-shape is called the T-number.
The T-number for this T-shape is 20.
1) The relationship between the T-Total and the T-Number.
If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.
T-total = T-19+T-18+T-17+T-9+T
= 5T-63
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
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 24+25+26+34+43=152
X=215-63
X=152
T=49
X=5x49-63 30+31+32+40+49=182
X=245-63=182
I have tested the formula for three different T-numbers and the formula works for a 9 by 9 grid.
The full formula for this size grid is: X=5T-63
I will now try to find out the formula of a 6 by 6 grid
If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.
= 5T-42
1) Now to test this formula to see if it works, again
For the T-total I will use the letter X
For the T-Number I will use the letter T
So X = 5T-42
T = 14
X = 5x14-42
= 70-42
= 28
Now I will do two more to check to see if it will work anywhere on the grid.
2)T=22
X=5x22-42 9+10+11+16+22=68
=110-42
= 68
3) T=35
X=5x35-42 22+23+24+29+35=133
=175-42 =133
I have tested the formula for three different T-numbers and the formula works for a 6 by 6 grid.
The full formula for this size grid is:- X=5T-42
I can predict that that the formula for the 3 by 3 square will be 21. I will now prove this.
So X = 5T-19
T = 8
X = 5x8-19
= 40-21 = 19
The full formula= X=5T-21
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 i.e. G=9
The formula is X=5T-7G
Which changes to X=5T-7x9 X=5T-63.
Rotation of T shapes
Red T shape = T= 4
X= 90 = 5T-90
X= 5x4 – 90 = 20-90= -70
X= 5T-70
Green T shape = T= 17
X= 92 = 5T-92
X= 5x17 – 92 = 85-92= -7
X= 5T+7
Purple T shape
T= 68
X=347= 5T-90
X=5x68-347= 340-347= -7
X=5T-7
Blue T Shape
T=75
X= 368= 5T-368
X=5x75-368
375-368= 7=
X= 5T-7
I HAVE FOUND THAT IF THE T-NUMBER IS EVEN THE LAST NUMBER IN THE T-TOTAL ENDS IN 70 AND IF ODD IT ENDS IN 7.
I will now try a 9 by 9 grid
Red T Shape
T=21
X=42
X= 5T-42= 5 x 21-42= 105-42
5T-63
Green T Shape rotated 90%
T=21
X= 112
X= 5T-112= 5 x 21 = 105-112=
X= 5T+7
Findings: Excluding non rotation, I have found that In the 9 by 9 grid if the T-Number is even the last number in the T-Total is 7 and if odd the last number is 2. In the 6 by 6 grid I have found out that if the T-Number is even the last number in the T-Total is 8 and if odd the last number is 3. I have also found out by doing these grids that as well as having one rule for each grid to work out the T-totals possible with in it, there can also be overall rules to work out any T-total of any grid size be it a 4 by 4 grid or a 20 by 20 grid. I also worked out that you always multiply the T-number by 5 and to get the number which you have to subtract all you do is multiply the grid size by 7.Which is 5t-7g
Rotations only If you rotate the T shape 90 degrees the formulae is 5t+7 and if you rotate it 180 degrees the formulae is 5t+7G. Also if you rotate it 270 degrees the formulae is 5t-7.
Evaluation:
