T-number=20 T-number=21 T-number=22 T-number=23 T-number=24
T-total= 37 T-total=42 T-total=47 T-total=52 T-total=57
As the T-number increases by 1, the T-total increases by 5 as each number inside it increases by one in the T. To determine an algebraic equation to relate the value of the T-number to the T-total, a value has to be assigned to the T-number, for the purposes of this investigation I will use N. As each time the T-total increases by N, the equation for the relation is: 5N+X, where X represents the value that is added to the 5n to determine the T-total. Using the fact that where the T-number is 20 and the T-total is 37:
(5*20)+X=37
100+X=37
X=37-100
X=-63
5N-63=T-total
This equation can also be shown in algebra as such:
N+N-9+N-18+N-17+N-19
5N-63
This means that multiplying the T-number by 5 and subtracting 63 will result in the T-total. This is as 5 is the number added whenever 1 is added to a T-number, and 63 relates to the grid size as each of the numbers that is subtracted in the algebraic T is subtracting the grid size.
As there is a direct relation between the second number in the equation and the grid size, the next logical step is to change the grid size, to insert algebraic T’s and work out the values of these. The smallest grid size that this T will fit into is a 3x3 grid size, so this is the smallest that I will start with.
3x3 4x4 5x5
5N-21 5N-28 5N-35
6x6 7x7 8x8
5N-42 5N-49 5N-56
10x10
5N-70
This results table show that every time the grid size increases by one, the number subtracting the 5N is increased by 7. This means, if G is used to represent the grid size, finding out the term of the formula algebraically, you come out 5N-7G. To prove this in algebra, the following ‘T’ is used:
N+N-G+N-2G+N-2G-1+N-2G+1
5N-7G
This shows how the T-Total in a normal T in any grid size relates to the grid number. It also shows how there is no numerical addition or subtraction to the entire sum, as the 1’s in the equation cancel.
A numerical example to prove this is a random ‘T’ from a 6x6 grid:
35+29+23+22+24=133
(35*5)-(7*6)=133
So far this investigation has only focussed on the standard ‘T’ position, which is at 0°. The T could also be analysed at the 90° clockwise rotation, the 180° rotation and the 270° clockwise rotation. At any other rotation, the standard squares will not fit onto the grid.
To use the 90° rotation, the T will be displayed as such:
The arrow is shows the consecutive T’s I will evaluate.
T=10 T=11 T=12 T=13 T=14 T=15
TT=57 TT=62 TT=67 TT=72 TT=77 TT=82
As the T number increases by 1, the T total increases by 5, as each number within the T increases by one.
