Which will give me the following pattern:
T + 5
N + 1 ratio 1:5
Now we got this information so we can find the formula. I have found a formula which is
5N - 63 = T
I have worked this formula by T-Number subtracting The T total which would look like this:
(N-11) + (N-1) + (N-2) + (N-3) = T
Which is basically doing this:
20 – 11 = 9
20 – 1 =19
20 – 2 =18
20 - 3 = 17 +
Total = 63
I will now prove my formula is correct :
5 * 20 - 63 = 37 T
N
5 * 20 = 100 100 – 63 = 37
To prove this again I will move the T-shape to another place. I have called the T-shape with T-number 20 A and the T-shape with the T-number 23 B which are resultants of the vectors.
Vector AB=
5N - 63 = T
5 * 23 – 63 = 52 T
N
5 * 23 = 115 115 – 63 = 52
This has proven that my formula is correct.
In the next section I will be translating a different size of grid which will be
8 by 8 grid.
Which would give me the following results
I will find the formula as I did before
(N-10) + (N-1) + (N-2) + (N-3) = T
18-10= 8
18- 2=16
18- 1=17
18- 3=15 +
Total = 56
5N-56= T
5*18= 90 90-56=34
This has proven my formula is correct but lets check again by moving the T-shape I called the T-shape with the T-number 16 = A and the T-shape with the T-number 18=B
Vector AB=
5*20= 100 100-56=44
Yes, the formula is correct
As you see both grids that I have investigated are in the 7 times table and I predict that the following 7 by 7 grid will be 7*7= 49. This gives me the following formula
5N-(7*grid size)=T
Which will give me the following results
I will use the same method as before to find the formula.
(N-9) + (N-1) + (N-2) + (N-3) = T
16-9= 7
16-1=15
16-2=14
16-3=13 +
Total = 49
This shows me that my predictions were right as the T is 49 in the case of the 7by 7 grid
5N- (7*G)=T
5N - 49= T
5*16= 80 80-49= 31 T
N
My formula is correct
Lets check if my formula is correct by moving the T-shape to another place where I have called the T-shape with the T-number 16 = A and the T-shape with the T-number 18=B
Vector AB=
5*18= 90 90-49= 41 T
N
In this section I will investigate the relationship between the T-totals, the numbers, size grid and the transformations.
When rotate the T-Shape in the 9 by 9 grid 180° we cannot use the method of vectors as it is an reflection but if we were to move this shape diagonally or vertically without rotating the shape we would be able to use vectors to describe translations but as this is not the case we cannot do this
We need to change the – to a + because we turned it around so we use the same formula as before but change the – to + so 5N+(7*G)=T which is 5N+63= T
(N-11) + (N-20) + (N-19) + (N-21) = T
2-11= -9
2-20=-18
2-19=-17
2-21=-19 +
Total = - 63
Change – to +
5*2+63= 73
5N+ (7*G)=T
T= 2+11+19+20+21= 73
Our formula is correct and reversing the minus has worked.
The next step will be to turn the T-Shape on it’s side which would look like this
To find the formula we use the same method as before.
(N-1) + (N-10) + (N-19) + (N-11) = T
12-1=11
12-10=2
12-19=-7
12-11=1 +
Total= 7
Our formula will be
5N-7=T
12*5-7=53
Lets check now if our formula is correct
T-total= 1+10+19+11+12= 53
Our formula is now proven to be correct.
If we now rotate the shape 180 degrees from the previous shape the same has to happen as we did before so here we have to change the minus to plus as well.
(N-17) + (N-18) + (N-27) + (N-9) = T
16-17= -1
16-18= -2
16-27= -11
16-9 = 7 +
-7 change – to +
5N+7=T
5*16=80 80+7=87
Lets see if this works
T=16+17+18+27+9=87
our formula is proven to be correct
I am now going to see if we can put the T-shape diagonally and still use the same method.
(N-29) + (N-12) + (N-31) + (N-2) = T
46-29= 17
46-12= 34
46-31= 15
46- 2= 44 +
110
5N-110=T
5*46=230 230-110=120
Lets check if our formula is correct
T=46+29+12+31+2=120 our formula is correct
Going back to what I previous said about skipping one place I have come up with a prediction which is that we could use a different formula for example 10N – 63 = T for the 9 by 9 grid which is 10*20=200 but this already shows us that the answer is 137 which proves my predictions wrong as it is just a pattern that it follows although there is a small link of 100 being added to the t-total this is also a sequences that it follows you can see it by doing the same procedure with the next T-Numbers
As you see above every time 100+10 is added to the T-total when you using this formula you can also see that T-number 21, 22, 23, 24 of the formula 10N-63=T is different as it does not follow the pattern of the formula as you see t-number 21 of the f 5N-63=T is does not get 142 but 147 because it skips one place as we are using the formula 10N-63 = T.
In conclusion I have found out different ways to translate and solve T-shapes in different positions with different grid sizes. I have solved the T-shapes by using a formula which slightly changes in different circumstances the formula is 5N-(7*G)=T which has linked the relationship between the T-Total, T-Number and the Grid size.
