N.B- The section of grid 9 below does not show the general formulae for the grid, it shows the formulae if the T- number was placed in the position that it is in now. Therefore if the T-number was placed in a different position than the formulae in the grid will change accordinglly.
(Fig.1.3) Section from 9x9 grid
Therefore the general structure of the T-shape is shown in fig.1.4:
(Fig.1.4)
Using the general structure of the T-shape (shown in Fig.1.4) I can now find a general formula to find the T-total using the numbers inside the T-shape:
T-total = T+(T-9)+(T-19)+(T-18)+(T-17)
= 5T-63
Therefore the general formula for finding the T-total for any T-shape in a 9x9 grid is:
5t- 63
I will now test my general formula to prove that it is correct.
Fig.1.5 shows five different T-shapes.
(Fig.1.5)
Fig.1.6 shows the T-totals worked out by adding all of the numbers in the T-shape together.
1+2+3+11+20= 37
13+14+15+23+32= 97
7+8+9+17+26= 67
48+49+50+58+67= 272
52+53+54+62+71= 292
(Fig.1.6)
Fig.1.7 the T-totals of the T-shapes in Fig.1.5 worked out using the general formula.
(Fig.1.7)
Comparing figures 1.6 and 1.7, I noticed that the T-totals are the same. This proves that my general formula is correct.
QUESTION 2
In Fig.2.1 I have positioned the T-number in different places on the grid. The table in Fig.2.2 shows the T-numbers from Fig.2.1 and their T-totals. I aim to find a connection between them.
(Fig.2.1)
(Fig.2.2)
After choosing the T-numbers and working out their T-totals, I found that as the T-number increases in size the T-total also increases in size with it.
I aim to find a general formula linking the T-number and the T-total for other grid sizes and maybe one single formulae for all the grid sizes together and not just one like the 5T- 63 for the 9 x 9 grid.
(Fig.2.3)
From looking at Fig.2.3 I noticed that the numbers 17,18,19 etc. can be written as a single formula. I will now have to introduce “g” to represent the grid size because this time I will be changing this. In quesion 1 I noticed when finding the general formulae for the 9x9 grid that the number at the end of the formulae is 63. This can be worked out by multiplying the number 7 with 9 which is the grid size. I will now try this with another grid size, 5:
(Fig.2.4)
Fig. 2.4 shows the grid of 5x5 and I have randomly picked two different T-shapes from the grid and highlighted them. The T-numbers are 12 and 18 and in Fig.2.5 I will test my formula with these two T-shapes.
1+2+3+7+12 = 25
5x12 – 5x7 = 25
7+8+9+13+18 = 55
5x18 – 5x7 = 55
(Fig.2.5)
Fig.2.5 shows that my formula works for the 5x5 grid. I can now change the number that represents the grid size and change it to just one letter, G, which can be used for all grid sizes. Therefore one can find the T-total of any ant T-shape that is in any grid size using the following formula:
T= 5t-7n
I will now prove that my formula works for a few more grid sizes before confirming that it is definately correct. Fig.2.6 shows these tests: