N
We can test this out with the T-shape: 53 54 55
62
70
T=5(70)-56
T=350-56
T=294
On a 7 x 7 grid
On a 7x 7 grid the formula is: T=N+N-15+N-14+N-13+N-7
T=5N-15-34
T=5N-49
Testing this with the T-shape: 37 38 39
45
52
we get T=5(52)-49
T=260-49
T=211
By these results we discover a sequence and that sequence is: All formulas for grid sizes have 5N(t-number)-a multiple of 7
To prove this we use the grid size 5 x 5.
If we work out the T-Total we get 30. The formula is then:
T=N+N-11+N-10+N-9+N-5
T=5N-35
If we look at 35 (called x) we notice again it is a multiple of 7 like 49 of the 7 x 7 grid and 63 of the 9 x 9 grid.
If we create a table we can notice a pattern.
From the table of a T-number based on its relationship between N(T-number) and g(grid size) we find: N-(2g+1) N-2g N-(2g-1)
N-g
G
To prove this we use the grid size 3 x 3.
8-(2(3)+1) 8-2(3) 8-2(3)-1)
8-3
8
From this we generate the formula: T=N-(2g+1)+N-(2g-1)+N-2g+N-g+N
T=5N-7g
We have taken: 5N (the number of numbers within the T-shape by the T-number)
7g (x are all multiples of 7)
3). Relationship between the T-total, the T-number and the grid size and other transformations using different grid size.
Rotations
To start with I must find out the rule of the rotations where the middle number is the centre of rotation on a 9 x 9 grid. In this case the centre of rotation will be called c.
The T-shape has a total of 52 but when rotated 90° the total is 72.
If we rotate the shape 180°and 270° we find some more generalisations and create a table.
So when rotated the T-total is longer, to prove this we use a different size grid.
On a 7 x 7 grid.
To find the formula we must break the results down.
180°
As the T is upside down the equation is easy to find.
The formula is 5c+2g
We get this by: C-7
C
C+6 C+7 C+8
T=C+C-7+C+6+C+7+C+8
T=5N+14
T=5N+(14divided by 7) (the sum is divided by 7 because this is the grid size)
T=5N+2g
To prove this we use a grid size 4 x 4.
3+7+10+11+12=43
Through these numbers we find the formulas
T=(5x7)+(2x4)
To double check we use the grid size 5 x 5
2+7+11+12+13=45
T=(5x7)+(2x5)
This proves that T=5c+2g can be used to find the T-shape at 180°.
90° and 270°
3+4+5+6+9=27
The formula can be found from: c-2
c-1 c c+1
c+4
Therefore if we substitute c for numbers we get:
T=(5+1)+(5+4)+(5-2)+(5-1)+5
T=27
Therefore we find the formula T=(c-1)+c+(c+1)+(c-2)+(c+4)
T=5c+2
To prove the formula we use a 4 x4 grid:
3+7+5+6+11=32
T=5c+2
T=(5x6)+2
T=30+2
T=32
This proves that the formula T=5c+2 can be used to find the T-total of any 270° shape where c is the centre of rotation.
The relationship between the 90° and the 270°is very similar so for a 90° flip the formula must be T=5c-2.
To prove this we use the grid size 6 x 6.
3+9+15+10+11=48
When we use the formula:
T=5c-2
T=(5x10)-2
T=50-2
T=48
Therefore proving that the formula is T=5c-2 for a 90° flip and can find the T-total
Now we can form the table:
We can also find different formulas.
180°
To find this formula we use the 3 x 3 grid.
2+5+7+8+9=31
We can find the T-total through: N
N+3
N+5 N+6 N+7
T=2+(2+3)+(2+5)+(2+6)+(2+7)
T=31
Therefore we discover the formula.
T=N+(N+3)+(N+5)+(N+7)+(N+6)
T=5N+21
T=5N+7g
This formula is also a T-shape formula turned around. To prove this we use the grid size 4 x 4.
2+6+9+10+11=38
T=5N+7g
T=5(2)+7(4)
T=10+28
T=38
Therefore proving that we can use the formula T=5N+7g to find the total of any grid size.
90° and 270°
To find the formula for this we use grid size 4 x 4.
3+5+6+7+11=32
N-2
N N+1 N+2
N+6
T=N+(N+1)+(N-2)+(N+2)+(N+6)
T=5N+7
To prove this we use the grid size 7 x 7.
5+10+11+12+19=57
By using the formula we find: T=5N+7
T=5(10)+7
T=50+7
T=57
We get the T-total.
By this we know that by using the formula T=5N+7 we can the T-total on any grid size.
To find the T-total of a 90° flip we simply turn the equation around giving us T=5N-7.
To prove this we us the grid size 5 x 5.
2+7+12+8+9=38
Using the formula: T=5N-7
T=5(9)-7
T=45-7
T=38
Thus proving the formula T=5N-7 can be used on any grid size to find the T-total. Now we can create a table:
45° and 225°
The rule for a diagonal shape is much the same as a straight shape. First I will find the relationship on a 9 x 9 grid.
45°
19+29+39+21+13=121
The difference in each T-shape is N
N+6 N+8
N+16
C+26
T=N+N+6+N+8+N+16+N+26
T=5N+56
On a 7 x 7 grid the formula is:
8+16+10+4+24=62
The difference is: N
N+4 N+6
N+12
N+20
T=N+N+2+N+6+N+12+N+20
T=5N+42
The overall formula of a 45° is T=5N+(7g-7)
To prove this we use grid six 10 x 10:
11+22+33+13+4=83
T=5N+(7g-7)
T=20+(70-7)
T=20+63
T=83
This proves that the formula for a 45° angle is T=5N+(7g-7)
225°
To find the formula for this we will start with a 9 x 9 grid:
7+17+27+25+33=109
The difference is: N-26
N-16
N-8 N-6
N
T=N+N-8+N-16+N-26+N-6
T=5N-56
On a 7 x 7 grid the formula is:
5+13+21+19+25=83
The difference in each T-shape is: N-20
N-12
N-6 N-4
N
T=N-20+N-12+N-4+N-6+N
T=5N-42
The overall formula for a 225° is T=5N-(7g-7)
To prove this we use the grid size 10 x 10:
8+19+30+28+37=122
T=5N-(7g-7)
T=185-(70-7)
T=185-63
T=122
This proves that the formula for a 225° is T=5N-(7g-7)
135° and 315°
135°
First I will find the relationship on a 9 x 9 grid:
19+11+3+21+31=85
The difference is: N-28
N-20
N-12 N-10
N
T=N-28+N-20+N-12+N-10+N
T=5N-70
On a 7 x 7grid the formula is:
3+9+15+17+25=69
The difference is: N-22
N-16
N-10 N-8
N
T=N-22+N-16+N-10+N-8+N
T=5N-56
The overall formula for a 135° angle is T=5N-(7g+7)
To prove this we use rid size 10 x 10:
21+12+3+23+34=93
T=5N-(7g+7)
T=170-(70+7)
T=170-77
T=93
This proves that the formula for 135° T-shape is T=5N-(7g+7)
315°
First I will find the relationship on a 9 x 9 grid.
5+15+25+17+33=95
The difference is: N
N+10 N+12
N+20
N+28
T=N+N+28+N+20+N+12+N+10
T=5N+70
On a 7 x 7 grid the formula is:
4+12+14+20+26=76
The difference is: N
N+8 N+10
N+16
N+22
T=N+N+8+N+10+N+16+N+22
T=5N+56
The overall formula for a 315° angle is T=5N+(7g+7)
To prove this we use the grid size 10 x 10:
5+16+27+18+36=102
T=5N+(7g+7)
T=25+(70+7)
T=25+77
T=102
Now we can create a table of results for rotation:
Larger T
Now that we have worked out the formulas of normal T’s we enlarge the shape. If we double the T-shape so the volume is 4 times bigger the grid shows the new shape.
1+2+3+4+5+6+9+10+11+12+13+14+19+20+27+28+35+36+43+44=342
The T-number is 158 as the bottom 4 numbers make up the T-number and the T-total is 342.
The difference in each number (the 4 groups of numbers) is: N-136 N-128 N-120
N-64
N
T=N+N-64+N-136+N-120+N-128
T=5N-448
To prove this we use:
T=5N-448
T=6x158-448
T=342
Now we must find the overall formula. We now use a 12 x 12 grid:
1+2+3+4+5+6+13+14+15+16+17+18+27+28+39+40+51+52+63+64=478
The T number is 230 and the T-total 478. The difference in the T-shape is:
N-200 N-192 N-184
N-96
N
T=N+N-96+N-192+N-184+N-200
T=5n-672
To prove this:
T=5N-672
T=1150-672
T=478
To find the overall formula we must make a table of results
With this table we notice a pattern and that is that every the difference in each is 56.
The formula can be easily formed:
T=5N-(7gx8)
To prove this we use a 15 x 15 grid:
1+2+3+4+5+6+16+17+18+19+20+21+33+34+48+49+63+64+78+79=580
T=5N-(7gx8)
T=1420-(105x8)
T=1420-(840)
T=580
This proves that our formula is correct.
