Check: 34 + 35 + 36 + 44 + 53 = 202
Now we can be 100% sure that the formula for finding out the T-Total in a grid with 9 numbers in a row is 5*T-Number-63.
I also found another formula of finding out the T-Total. This formula would be:
T = N + (N - 9) + (N - 17) + (N - 18) + (N - 19)
This formula can be used, too because the distance between the numbers in a
T-Shape never changes. For example:
So in this case the equation would be:
T = 50 + (50 – 9) + (50 – 17) + (50 – 18) + (50 – 19)
T = 50 + 41 + 33 + 32 + 31
T = 187
Check: 5*50-63 = 250 – 63 = 187
The following shows a T-Shape with the used formula to explain what the formula for each number is when we have the T-Number.
PART 2
This next section involves using grids of different sizes and then translating the T-Shape to different positions. After this I will have to investigate 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.
I am going to use the grid with eleven numbers in a row:
We will start again with the T-Shape with the lowest T-Number, which will be 24. The T-Total would be 1 + 2 + 3 + 13 + 24 = 43
We can again use the longer formula as in Part 1, which would be:
T = N + (N – 11) + (N – 21) + (N – 22) + (N – 23)
To prove that this formula is correct I am going to use this for the already known T-Number 24:
T = 24 + (24 – 11) + (24 – 21) + (24 – 22) + (24 – 23)
T = 23 + 14 + 3 + 2 + 1
T = 43
Comparing to the first grid (9x9) the T-Total and the T-Number in this grid (11x9) 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. If we use the same rules we made in the last section we can find out the formula for the T-Total easily.
First we need some results to compare between them more easily.
1 + 2 + 3 + 13 + 24 = 43
2 + 3 + 4 + 14 + 25 = 48
3 + 4 + 5 + 15 + 26 = 53
4 + 5 + 6 + 16 + 27 = 58
We can see again that if the T-Number goes one up the T-Total goes five up. This is because by raising the T-Number by one, each number of the T-Shape will go one up. Because there are five numbers in a T-Shape the T-Total goes 5*1 = 5 steps up.
This means again that the formula will have to start with 5*T-Number and end with a subtraction of a number that we will find now.
To find this subtrahend we have to methods. I will start with adding the differences between the T-Number and the four other numbers in the T-Shape (T-Number is 24):
(24 – 13) + (24 – 3) + (24 – 2) + (24 – 1)
= 11 + 21 + 22 + 23
= 77
The second method would be as followed (T-Number is 24):
5*24 – 43 = 77
I got the number 43 by adding the numbers of the T-Shape with the T-Number 24. For this I just looked up to the results.
Now I am going to test this formula.
The T-Number 98
98*5-77 = 413
Check: 75 + 76 + 77 + 87 + 98 = 413
The T-Number 32
32*5-77 = 83
Check: 9 + 10 + 11 + 21 + 32 = 38
The T-Number 70
70*5 – 77 = 273
Check: 47 + 48 + 49 + 59 + 70 = 273
The T-Number 61
61*5 - 77 = 228
Check: 38 + 39 + 40 + 50 + 61 = 228
Next I am going to use one more grid, which will be smaller than the first one. I will use a grid with only 4 numbers in a row.
I will start again with the T-Shape that has the smallest T-Number. In this grid it will be the T-Number 10. The T-Total for this T-Shape will be:
1 + 2 + 3 + 6 + 10 = 22
The formula for this shape will again start with 5*T-Number because the T-Shape still has five numbers and if each number goes one up the T-Total will have to go five up. The last number I will again find out by calculating the followings:
5*10 = 50
50 – 22 = 28
Therefore the formula has to be 5*T-Number – 28.
To check whether this formula is correct I will need some more results.
T-Number 11
11*5-28 = 27
Check: 2 + 3 + 4 + 7 + 11 = 27
T-Number 18
18*5-28 = 62
Check: 9 + 10 + 11 + 14 + 18 = 62
After finding out the formula for each of the grids I looked at the formulae. These were:
9 5* N - 63
11 5* N -77
4 5* N -28
By looking at these formulae I saw that always the last number is a factor of 7. By dividing each number by 7 you get the amount of numbers in each line.
63/7 = 9
77/7 = 11
28/7 = 4
Therefore the general formula for finding the T-Total by using the T-Number is:
T = 5*N – (G*7)
E.G. the formula for the T-Total for a grid with 9 columns will be:
T = 5*N – (9*7)
T = 5*N – 63
There you can see that this is the same formula as I wrote and proved for the first grid, which had 9 columns.
PART 3
180° rotated T-Shape
In this section there is change in the size of grid. There is also transformation and combinations of transformations. I need to find out the investigation of the relationship between the T-Total, the T-Number, the grid size and the transformations.
If I turned the T-Shape around 180 degrees it would look like this.
Once I have done this I realized that if I reverse the T-Shape I should have to reverse something in the formula.
The formula has a multiplication and a subtraction sign in it and that one of these two has to be turned into the negative. It cannot be the multiplication because in this case the T-Total would be smaller than the T-Number, whereas it is not possible to have a smaller T-Total than the T-Number. Therefore I think I have to turn the subtraction sign into an addition sign. After this the formula should be:
5*N – 63 → 5*N + 63
Now I need to check this formula.
The T-Number will be the number at the top of the T-Shape.
The formula would be:
5*2+63 = 73
Check: 2 + 11 + 19 + 20 + 21 = 73
The formula has worked.
A general formula for finding the T-Total in a 90° clockwise rotated shape this would be:
5*N + 7G
This is because the last number occurs by multiplying the grid size by seven.
270° rotated T-Shape
Next I will find out the formula for a 270° clockwise rotated T-Shape.
We need to add up all the differences between the T-Number and all the other numbers in the T-Shape. This would be:
12 – 11 = 1
12 – 10 = 2
12 – 1 = 11
12 – 19 = -7
1 + 2 + 11 – 7 = 7
The result of adding up all these differences is 7. This means the formula for a T-Shape, which is 270° clockwise rotated, is 5*N – 7
To prove this I am going to check it on the following shape:
5*25 – 7 = 118
Check: 14 + 23 + 32 + 24 + 25 = 118
This proves that this formula is correct for this grid size. For another grid size we just need to add up all the differences between the T-Number and the other numbers in the T-Shape and use the answer as the subtrahend.
Translation
Next I am going to show the correlation between a normal T-Shape and a translation of that one.
Here you can see a grid with nine columns and the T-Shape, which has been translated to the new one.
This is a translation of (2 | -4). We already know that the T-Total for the first shape is 37 (5*20-63 = 37). Now we need to find out the T-Total for the new shape, which would be
39 + 40 + 41 + 49 + 58 = 227
Next I am going to find out a formula for finding the T-Total of the new shape.
I know that the translation is (2 | -4). This means that the new Shape has moved two steps to the right and four steps downwards. So I need to find out how to get from the first T-Number to the second T-Number. For this I have to add 2, because the new Shape moves 2 steps to the right and add 4*9, because I go 4 steps down and each step down increases the number by 9. After this I just need to add the general formula for finding a T-Total, which is …*5-63 (for a grid with nine columns.
So the final formula for finding the T-Total of the new T-Shape will be:
(20+2) + (4*9)*5-63 = 227
This shows that the formula is correct because it gives the same result (227) as the calculation (s. above).
My next step will be to find a general formula for a translation of (x | y). For this I will first demonstrate what the formula for each space is:
By looking at these results we can also find another formula, which is the sum of the following formulae:
N + x – yG – 2G – 1
N + x – yG – 2g +1
N + x – yG – 2G
N + x – yG – G
N + x – yG
= (N + x – yG)
This is the formula for finding the new T-Number. But for finding the new T-Total we need to add *5-7G to the expression. The final formula is therefore:
(N + x – yG)*5 – 7G
To prove that this formula is correct I am going to use it in a table of another grid size. I will check the formula on a grid with twelve columns.
Here we can see a translation of (6 | 3). The calculations I now have to do is:
(53 + 6 + 3*12)*5 – (7*12)
= (59 + 36)*5 – 84
= 95*5 – 84
= 391
Check: 70 + 71 + 72 + 83 + 95 = 391
This means that this formula is correct.
Rotations
90° rotation
Now I will try to find out a formula for finding the T-Total of a new shape, which is rotated 90° clockwise, when you are given only the T-Number of the original T-Shape.
As you can see on the grid, the T-Shape is rotated on the point 31 by 90° clockwise. To find the formula we first need to find the difference between the original T-Number and the rotation point. In this case this is (2 | -1). The difference from the rotation point to the new T-Number is (1 | -2). You can see that both differences have the same figures. They have just changed the place and they were multiplied by -1. Now you can easily create a formula:
20 +2 +(1*9) + 1 – (2*9) = 14
Now I will explain where these numbers come from:
- 20: the original T-Number
- + 2: moving two steps to the right
- +(1*9): moving one step downwards (9 – grid size)
- +1: moving one step to the right
- - (2*9): moving twice the amount of numbers in a row upwards
After finding this calculation I need to find a general rule, which works on every grid size and every rotation.
In this case I am going to use the factors a and b because x and y were used before in translation.
I know this rule will start with N + a – b, because by doing this you get from the original T-Number to the rotation point. After this you need to multiply the a-value and the b-value by -1 and change their places. So the formula would become y – x instead of x – y. A sensible formula would be (N + a – bG) + (b – aG). This formula leads you to the new T-Number.
To check that this formula is correct I am going to show this in a grid. Here the T-Shape has been rotated on the point 67:
Now I am going to use the formula and put in the original numbers.
(N + a – bG) + (b - aG)
= (38 + 2 + 3*9) + (3 – 2*9)
= 40 + 27 + 3 – 18
= 67 – 15
= 52
With this formula we get to the new T-Number in a 90° rotation. Next we need to find and add the formula for a 90° clockwise rotated shape.
We can find this formula easily by doing followings:
We need to add up each formula in this T-Shape above.
N
N + 1
N + 2
N + 2 – 2G
+ N + 2 + 2G
_ .
= 5N + 7
So the final formula for a 90° clockwise rotation is
5(N + a – bG) + (b - aG) + 7
Conclusion
In this project we have found out many ways in which we can solve a problem that we might have with the T-Shape. It can be in various different positions with different grid sizes. We made the calculation easier by creating a main formula, which can change for all the different circumstances.
The followings are all the formulae I found out and proved in this project. They can work for any grid size.
Normal T-Shape 5N - 7G
90° clockwise rotated 5N + 7
180° clockwise rotated 5N + 7G
270° clockwise rotated 5N - 7G
Translation of T-Shape (N + x – yG)*5 – 7G
90° rotation 5(N + a – bG) + (b - aG) + 7
Page
T = T-Total; N = T-Number; G = Grid size;
x = horizontal movement; y = vertical movement (for translation)
