Example:
The T-shape has a T-number of 32. So now you work out the difference between the T-number and the remaining numbers in the T-shape:
Working out:-
32 – 13 = 19
32 – 14 = 18
32 – 15 = 17
32 – 23 = 9
Total = 63
This will happen to all the shapes this way up.
To prove it I will do another example:-
The T-Number is 70. So now we work out the difference between the T-number and the rest of the numbers once more:
Working out:-
70 – 51 = 19
70 = 52 = 18
70 – 53 = 17
70 – 61 = 9
Total = 63
Again the number turns out to be 63. this is where the 63 came from in the equation. There is also another way the 63 comes in. this is 9x7=63. the nine comes from the size of the grid(9x9). If the grid was 10x10 then it would be 10x7. The numbers we plus or minus is divisible by 7. this is where we get the seven from.
If we add these two together we have a formula:-
5T-63=T-Total
This is how the Formula will work:-
5T-63 = T-Total
5x57-63 = T-Total
5x57-63 = 222
To check if it is correct we add all the numbers in the T-shape together
T-Total = 38+39+40+48+57 = 222
Next I am going to use grids of different sizes to further carry investigation. I am going to investigate the relationship between T-number, T-total and grid size.
T-Total = 1+2+3+13+24 = 43
T-Number = 24
The T-total and T-shape have risen even though the T-shape is still in the same place as the 9x9 grid. The T-number has risen by four and the T-total has risen by six. If we do the same as before it still works. There are two methods. Long and short.
Difference
Long method
24-1 = 23
24-2 = 22
24-3 = 21
24-13 = 11
Total = 77
Or the short way
7 x 11(grid size) = 77
5T – 77 = T-Total
5 x 24 – 77 = 43
In this section I will again use grids of different sizes, Transformations and Combinations of Transformations. I will investigate relationships between the T-total, the T-numbers, the grid size and the transformations.
If we turned the T-shape around 180 degrees it would look like this:-
When we have done this we should realise if we reverse the T-shape then we should have to reverse something in the formula.
It is quite obvious that we would have to change the minus to a different sign. I will try the opposite of minus which is plus.
5T +63 = T-total
5 x 2 + 63 = 73
Now to check If the formula has worked
T-Number = 2
T-total = 2+11+19+20+21 = 73
The reverse in the minus has worked.
For the next step I will turn the T-shape on its side. We have to change the minus sign again. We can work out the number to minus by working out the difference in the T-number to each number in the T-shape
Difference
12-1 = 11
12-10 = 2
12-19 = -7
12-11 = 1
Total = 7
Formula
5T-7 = T-total
5x12-7 = 53
Check
T-number = 12
T-Total = 1+10+19+11+12 = 53
If the T-shape was rotated 180 degrees, the same will happen as what happened when the t-shape was turned 180 degrees from its first original position.
5T + 7 = T-Total
5x70 + 7 = 357
Check
T-Number = 70
T-Total = 70+71+72+63+81 = 357
If we were to put the T-shape diagonally on the grid we find that the same rule applies gain apart from you cannot use the 2nd rule where you multiply the grid size by seven.
T-number = 33
T-Total = 7+17+26+25+33 = 109
Difference
33-25 = 8
33-7 = 26
33-17 = 16
33-27 = 6
Total = 56
5T+56 = T-total
5x33-56 = 109
the reverse shape should be changed to a plus.
T-Number = 13
T-Total = 19+29+39+21+13 = 121
5T + 56 = T-total
5x13+56 = 121
T-Number = 32
T-total = 32+42+52+60+44 = 230
Formula
5T+70 = T-total
Difference
42-32 = 10
52-32 = 20
60-32 = 28
44-32= 12
TOTAL = 70
5 x 32 + 70 = 230
If you reverse this procedure however the Formula now will be 5T-70
32-42 = -10
32-52 = -20
32-60 = -28
32-44 = -12
TOTAL = -70
5 x 32 -70 = 90
As we can see from this table the difference between the t-number and the T-total goes up up four each time. So I have decided to put the difference in Algebraic terms: -
A = T-number
A + 17 = 37
A + 21 = 42
A + 25 = 47
A + 29 = 52
A + 33 = 57
From these graphs and tables I think it is fair to say that the higher the T-number the greater the T-total will be.
