To begin with I will use the T-Shapes in an upright position, & then as I get further into the investigation, I will start to rotate them into different positions, to see if my findings work in all positions rather than just one particular spot.
All of my results will be shown underneath the relevant grids with the T-Totals highlighted. All of the explanations will appear below the grids too.
With the results from the various grids, I hope to see an algebraic pattern forming.
Grid One – 9 x 9
The T-Number is 20 & the T-Total is 37
Grid Two 9 x 9
The T-Number is 21 & the T-Total is 42.
As you can see from the first two grids, the T-Number moves a square along, the T-Total goes up by five. This means that the ratio between the T-Number & the T-Total is 1:5.
This is useful for when you are changing the position of the T-Shape. For example when you move it up 5 places to 6, it will look like this;
The T-Number is 25 & the T-Total is 54. Another way to work out the numbers in green, besides adding them together is to work out the difference between the T-Numbers, which for this example is 54.
If you then multiply the 54 by 5, as the number rises by 5 every time the T-Number goes up. Then add the T-Total from the original shape to get the new T-Total number for the green shape.
This relationship between the T-Totals & T-Numbers can be formed together to make an algebraic formula.
5tn – 63 = T-Total. (tn = T-Number)
As the formula starts with 5 x the T-Number, because I have found that there is a rise of each time in the grid for every T-Number. Then I subtracted the 63, which I got by working out the difference between the T-Number & another number within the T-Shape.
This has to be done to all four numbers in the T-Shape in order for the formula to work correctly. An example of which is below in yellow.
The yellow T-Shape has a T-Number of 32. My workings out for the T-Number & the rest of the numbers in this T-Shape are as follows;
32-13=19, 32-14=18, 32-15=17, 32-23=9 Total=63
To prove that my findings are correct, I have drawn out another T-Shape in lilac on the above grid and followed the same rule as above. My workings for this second theory are below;
70-51=19, 70-52=18, 70-53=17, 70-61=9 Total=63
As you can see from both sets of results the final number is 63. This is also where the 63 in the equation came from. Another place where the 63 would originate from is the size of the grid, 7x9 = 63. However, if the grid size was 6 x 6, then it would be 6 x 9.
When all of the formulae are put together, the numbers that I used to plus or minus by are actually divisible by 7.
Therefore when all of the formulae are added together you get the following formula; 5tn-63=T-Total.
To prove that this formula has worked, I have provided an example below;
Workings;
5 x 57 – 63 = T-Total
5 x 57 – 63 = 222
Checking the theory is correct;
T-Total = 38 + 39 + 40 + 48 + 57 = 222
Now I will try to use the formulae from above in different sized grids, where I will also move the T-Shape to different positions within the grid.
I will then investigate the relationships between the T-Total, the T-Number & the size of the grid.
This time I have used a larger grid to see if there are any differences with the formula when a larger gird is used.
T-Total = 1 + 2 + 3 + 13 + 24 = 43 T-Number = 24
Even though the T-Number & the T-Total has risen the T-shape is in its original position. The T-Number has risen by 4 & the T-Total has risen by six.
If I use the same rule as before I get the following results, albeit a longer method.
The difference is as follows;
24 -1 = 23, 24 -2 = 22, 24 -3 = 21, 24 – 13 = 11 Total of 77.
However, if I did it the shorter way round using the formula, I would end up with this set of results;
5tn – 77 = T-Total7 x 11 (grid size) = 77)
5 x 24 – 77 = 43
In order to prove that this theory not only works on a larger grid of 11 x 11, I will try it on a much smaller grid of 4 x 4.
T-Total = 1 + 2 + 3 + 6 + 10 = 22 T-Number = 10
5tn – 28 = T-Total (grid size) = 28)
5 x 10 – 28 = 22
As you can see from the above example, by changing the grid size, I have not changed the formula slightly, but I have still managed to keep the minus rule.
Now, I will investigate the relationships between the T-Total, the T-Number, the grid size & the position of the T-Shape.
In the next grid I have turned the T-Shape around 180 degrees. As I have done this, I will also need to change the formulas slightly to compensate for this. I will do this by supplementing the minus sign for a plus sign, which is the opposite.
5tn + 63 = T-Total 5 x 2 +63 = 73
Check that the formula has worked by changing the signs;
T-Number = 2
T-Total= 2 + 11 + 19 +20 + 21 = 73
This example shows that I was correct to change the formulas around slightly.
By moving the T-Shape 180 degrees will not entirely prove that the theory is correct, so I will have to move it again to another position.
Here you can see that I have rotated the T- Shape around on its side. The formula for this shape will be similar to the one above by using the minus to calculate the differences in the T-Number to the other numbers in the T-Shape.
Calculation:
12 – 1 = 11, 12 – 10 = 2, 12- 19 = 7, 12 – 11 = 1 Total = 7
Formula:
5tn – 7 = T-Total 5 x 12 – 7 = 53
Check to see the formula works:
T- Number = 12
T-Total = 1 + 10 + 19 + 11 + 12 = 53
From this investigation of T-Totals, I conclude that there is a relationship between the T-Totals & the T-Numbers.
However, it all depends on where the shapes are placed & at what degree as the formulas have to be changed accordingly.
To investigate the relationships even further I could have perhaps tried them on a diagonal line or even increased the size of the T-Shape to see if the same rules applied here too.
