What we need now is a formula for the relationship between the t-total and the t-number
I will call the T-number “ n “
I will now draw the same T shape as shown below however this time using n as the t-number
If I put these numbers into a formula and add them up I will eventually get this
(n) + ( n-9) + (n-17) + (n-18) + (n-19) =
5n-63
I will now show how I got the 63 in the formula
The t-number is 70. Now to work out the difference between the t-number and the rest of the numbers in this t-shape
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 this equation. There is also another place this 63 comes from. This is 9 7=63. The nine in this comes from the size of the grid this one been nine. If the grid size were 10 by 10 then it would be 10 7. If we add these two together we have our formula.
5n-63=t-total
Here is an example of using the formula
5 57-63=t-total
5 57-63= 222
Check
T-total = 38+39+40+48+57=222
This formula has proven to work.
This next section involves using grids of different sizes and then translating the t-shape to different positions. Then investigation of the relationship between the t-total, the t-number and the grid size. Here we are doing what we did in the last section but finding out more about the grid size and what it is capable of doing.
This is an 11 by 11 grid
The t-total for the shape in red is
1+2+3+13+24 = 43
and the t-number is
T-number = 24
The t-total and the t-number have risen even though the t-shape looks to be in the same place. The t-number has risen by four and the t-total has risen by six. I will use the same method I used last time to fide out a formula for the 11 by 11 grid size:
(n) + (n-11) + (n-21)+ (n-22) + (n-23) =
5n - 77
I will show you how I got the 77 from the formula :
Difference
24-1= 23
24-2 = 22
24-3 =21
24-13 =11
TOTAL =77
The same formula works with only changing the last number in the formula. This will be tried on a smaller grid size
T-number = 10
T-total = 1+2+3+6+10= 22
This has proven to work on a smaller scale. We can see that by changing the grid size we have had to change the formula but still managing to keep to the rule of how you get the number to minus in the formula.
(n) + (n-4) + (n-7) + (n-8) + (n-9)=
5n-28
3.
In this next section there is change in the size of grid. Also there is transformations and combinations of transformations. The investigation of the relationship 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 we should have to reverse something in the formula.
It is obvious that we will have to change the minus sign to a different sign. We should try the opposite of minus which is plus
5n + 63=t-total
5 2 + 63 = 73
Check to see if the formula has worked
T-number = 2
T-total = 2+11+19+20+21 =73
The reverse in the minus sign has worked.
The next step is to move the shape on its side. Again we nearly keep the same formula as we had at the beginning. Again we change the minus number. 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
5n - 7 =t-total
5 12 - 7= 53
Check to see if the formula is right
T-number = 12
T-total = 1 +10 +19 +11 +12 = 53
This formula has worked. If we rotated the t-shape 180 degrees, The same will happen, as what happened when the t-shape was turned 180 degrees from it is first original position. This is proven below.
5n + 7 = t-total
5* 70 + 7 = 357
Check
T-number = 70
T-total = 70+71+72+63+81 = 357
I will now try and work out a formula for the grid size :
I will call the grid size “G”
(n) + (n-G) + (n-2G+1) + (n-2G) + (n-2G-1) = 5n-7G
I will now try out the formula on a 4 by 4 grid
- 10 = 50
50-7G
50-28=22
If I add up the number in the T you will find they add up to 22 so my formula works
If we were to put the t-shape diagonally on the grid we find that the same rule applies again apart from you can not use the 2nd rule were you times the grid size by seven.
The red t-shape has t-number of 33 and the t-total = 7+17+27+25+33 = 109
The difference between the t-number and the rest of the numbers in the t-shape.
33-25= 8
33-7= 26
33-17= 16
33- 27 = 6
TOTAL= 56
5n+56= t-total
5 33 - 56 =109
The reverse triangle the sign should be reversed to a plus. The t-shape used here is the one in blue.
T-number is 13
T-total = 19+29+39+21+13 = 121
5n+56= t-total
5 13+ 56= 121
If there are formulas for rotation then surly there is for reflection. Here I have simply only done one type of reflection just to prove that reflection actually works. Here is the formula 5n+ (12gm) = t-total. How do we get this formula is what we need to know.
The answer to this is that you need to think of what you are doing to each of the numbers in the t-shape from the blue t-shapes t-number. For the number 29 we have a grid movement of one so we get n ( t-number) + grid movement (gm)
(n+gm)
For the number 38 we have a grid movement of two so we get (n+2gm). For the numbers 46, 47 and 48 we have a grid movement of three and a total of three numbers, se we get 3(tn+3gm).
So ifi I put these into an equation I should get the formula above
(n+2gm) + 3(n+3gm) + (n+gm) =
formula for reflection = 5n+(12gm)
