The full formula for this size grid is X = 5T – 63
I then tried to find out the T-Total for 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.
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 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
X = 5T- 42
T = 14
X = 5 x 14 - 42
= 70 - 42
= 28
Now I will do two more to check to see if it will work anywhere on the grid.
T = 22
X = 5 x 22 - 42
9 + 10 + 11 + 16 + 22 = 68
= 110 - 42
= 68
T = 35
X = 5 x 35 - 42
22 + 23 + 24 + 29 + 35 = 133
= 175 - 42
= 133
To test the formula for three different T-numbers and the formula works for a 6 by 6 grid.
The rule or formula for this size grid is: X=5T-42
Below is a grid with parts of the formula in. I will use this to find 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 - 7 x 9
X = 5T - 63
I will now try the formula on a 7 by 7 grid.
X = 5 x 17 - 49 (2 + 3 + 4 + 10 + 17) = 36
X = 85 - 49
X = 36
I am now going to try this formula on a 3 by 5 grid.
X = 5 x 8 - 21 (1 + 2 + 3 + 5 + 8) = 19
X = 40 - 21
X = 19
I am now going to try this formula on a 6 by 10 grid.
X = 5 x 33 - 42 (20 + 21 + 22 + 27 + 33) = 123
X = 165 - 42
X = 123
Part 2
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.
A 3 by 3 grid can be used for the other 3.
T
T + G
T + 2G
T + 2G -1
T + 2G + 1
7G + 5 T
X = 7G + 5T = 7 x 3 + 5 x 2
= 21 + 10 (2 + 5 + 7 + 8 + 9 = 31)
= 31
I will now try this new formula on a 4 by 7 grid.
7G + 5T
X = 7 x 4 + 5 x 10 (17 + 18 + 19 + 14 + 10) = 78
= 28 + 50
= 78
I will now try it on a 8 by 3 grid.
7G + 5T
X = 7 x 9 + 5 x 2 (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: 7G + 5T
T
T-1
T-2
T-2-G
T-2 + G
5T - 7
X = 5T -7 = 5 x 6 - 7
= 30 – 7 (1 + 4 + 7 + 5 + 6) = 23
= 23
I will now try the new formula on a 4 by 4 grid
5T - 7
X = 5 x 11 -7 (5 + 9 + 13 + 10 + 11) = 48
= 48
On a 4 by 9 grid
5T - 7
X = 5 x 19 -7 (13 + 17 + 21 + 18 + 19) = 88
= 88
I have tested this new formula for a side ways T the formula works it is: 5T-7
This is the minimum t-shape possible that you can get on any grid size.
T
T + 1
T + 2
T + 2 + G
T + 2 - G
5t + 7
The formula is: 5T + 7
X = 5T + 7 = 5 x 4 + 7
= 20 +7 (3 + 6 + 9 + 5 + 4) = 27
= 27
I will now try the new formula on a 6 by 6 grid.
5T + 7
X = 5 x 15 +7 (11+ 17 + 23 + 16 + 15) = 82
= 82
And on a 11 by 5 grid.
5T + 7
X = 5 x 28 + 7 (28 + 29 + 30 + 19 + 41) = 147
= 147
I have tested this new formula for a side ways T the formula works it is: 5T+7
The formula for this way up is: X = 5T - 7G
The formula for this way up is: X = 7G + 5T
The formula for this way up is: X = 5T + 7
The formula for this way up is: X = 5T - 7
Part 3
I am now going to investigate the effect on the T-Total when the T-Shape is transformed, rotated 90°, both T-Shapes will have the same T-Number like this:
T- Number = 12
Purple T-Total = 25
Blue T-Total = 67 a difference of 42
The formula 5T – 7G + 42
So if my formula was right then for a 6 x 6 grid the answer would be
T- Number = 14
Purple T-Total = 33
Blue T-Total = 82 a difference of 49
The formula 5T – 7G + 49
I will not try out a 7 by 7 width Grid to see if my formula works
T- Number = 16
Purple T-Total = 41
Blue T-Total = 97 a difference of 56
The formula 5T – 7G + 56
Table of Results
Now I will rotate it again 180° from its original place
T- Number = 12
Purple T-Total = 25
Blue T-Total = 95 a of difference of 70
The formula 5T – 7G + 70
Now I’m going to see what happens on a 6 by 6 width Grid
T- Number = 14
Purple T = 28
Blue T = 112
A difference of 84 The formula 5T – 7G + 84
7 by 7 Grid
T- Number = 16
Purple T-Total = 31
Blue T-Total = 129 a difference of 98
The formula 5T – 7G + 98
Table of Results
Conclusion
From my analysis I have found that as long as the translation is the same the formula works, no matter if you enlarge the grid size or make it smaller. Also the number has to be a antigen number which you can multiply so that the formula works, if you rotate the T shape the formula will not work.
Sara Ali-Asghar GCSE Maths – Page of 2003