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Look at this t-shape drawn on
Thisbygrid.
The total of the numbers inside
Write my Essay on Maths T-Totals
The t-shape is 1+++11+0=7.
This is called the T-total.
The number at the bottom of the
T-shape is called the T-number.
The T-number for this T-shape is
0, the T-total for this T-shape
Is 7.
Part 1
In this part of my investigation I am going to investigate the relationship between the T-total and the T-number on thisbygrid.
1 4 5 6 7 8
10 11 1 1 14 15 16 17 18
1 0 1 4 5 6 7
80 1 4 5 6
7 840 41 4 4 44 45
46 47 48 4 50 51 5 5 54
55 56 57 58 5 60 61 6 6
64 65 66 67 68 6 70 71 7
7 74 75 76 77 78 7 80 81
Here to my left is a table of all the possible T-numbers and
There corresponding T-totals to abygrid.As you can see
Every time you mover the T-shape one space to the left it adds
5 onto the T-total.This is because every time you move it left
It adds 1 more number to each number in the T-shape.
So if I call the T-number T I can change the numbers in the
T-shape into formulas like this
1T-17 T-18 T-1
11SoT-
0T
You will notice that the centre column of the T-Shape is going
Up in 's because of the table Size.With the table set out like
This a formula Can be worked out to find any T-Total on this
Size Grid.This is done in the working below -
T-total = T-1+T-18+T-17+T-+T
So=5t-6
Now I will test it a few times to see if it works.
I will use x to represent the T-total and T to represent the
T-number.
So X = 5T-6
T = 0
X = 5x0-6
= 100-6
= 7
T=4
X=5x4-6 4+5+6+4+4=15
X=15-6
X=15
T=4
X=5x4-6 0+1++40+4=18
X=45-6
X=18
I have tested the formula for three different T-numbers and the formula works for abygrid.
The full formula for this size grid is - X=5T-6
Part
In partI will be using grids of different sizes.Translating the T-shape to different positions.And investigating the relationships between the T-total, the T-numbers and the grid size.
by
1T-5 T-6 T-7
5SoT-
8T
So the T-total = T-7+T-6+T-5+T-+T
So=5t-1
4 by 4
1T-7 T-8 T-
6SoT-4
10T
So the T-total = T-+T-8+T-7+T-4+T
So=5t-8
5 by 5
1T- T-10 T-11
7SoT-5
1T
So the T-total = T-11+T-10+T-+T-5+T
So=5t-5
6 by 6
1T-11 T-1 T-1
8SoT-6
14T
So the T-total = T-1+T-1+T-11+T-6+T
So=5t-4
7 by 7
1T-1 T-14 T-15
SoT-7
16T
So the T-total = T-15+T-14+T-1+T-7+T
So=5t-4
8 by 8
1T-15 T-16 T-17
10SoT-8
18T
So the T-total = T-17+T-16+T-15+T-8+T
So=5t-56
by
1T-17 T-18 T-1
11SoT-
0T
So the T-total = T-1+T-18+T-17+T-+T
So=5t-6
10 by 10
1T-1 T-0 T-1
1SoT-10
T
So the T-total = T-1+T-0+T-1+T-10+T
So=5t-70
11 by 11
1T-1 T- T-
1SoT-11
4T
So the T-total = T-+T-+T-1+T-11+T
So=5t-77
1 by 1
1T- T-4 T-5
14SoT-1
6T
So the T-total = T-5+T-4+T-+T-1+T
So=5t-84
So as you can see from the grid to the left every time
You increase the grid size by one the number that is
Subtracted at the end is increased by 7.
I you could have a grid size ofor 1 this would be
5T-14 forbyand 5T-7 for the 1 by 1.
This tells me that on a 1 by 1 sized grid it starts at 7
And then increases by another 7 to get 14 forby
Another 14 to get 1 forbyand so on.
Now that I have fully investigated grid sizes frombyright up to 1 by 1 I am going to see if I can make a formula for any grid size and to any translation.G=Grid size
T-G-1 T-G T-G+1
T-G
T
So now from this I can make a formula for that.
T-total = T-G-1
T-G
T-G+1
T-G
T
X=5T-7G
So I now have a formula for any grid size anywhere on the grid.
Part
Now I will have to find formulas for rotation about the T-number firstly on each grid separately then a formula that will make it able to rotate on any grid.
Proof For 0 degrees and 180 degrees
In the T-shape there are 5 squares.This is where I
Get the 5T from.For 0 shapes this is more than the T-number so I times the grid size by 7 and subtract this from 5T.But with 180 I have less than I need so this time I add the number I get when I times the grid size and 7 together.
Proof for 0 degrees and 70 degrees
As the same as 0 and 180 I get 5T from the number of squares in the T-shape.This number for 0 is now 7 less than I need so I add 7.For 70 it is 7 too much so I subtract 7 from it.
So now if I make a formula using G as like I did before
0=5T-(7G)
180=5T+(7G)
0=5T+7
70=5T-7
Part 4
For this part I will have to be able to translate it 'a' numbers across and 'b' numbers down and then I will have to be able to rotate it from that.
So now if I move it 'a' move across then this will happen
T-17 T-18 T-1T-17+a T-18+a T-1+a
T-=T-+a
TT+a
And then if I also incorporate the grid size as I did before
T-17+a T-18+a T-1+aT-g-1+a T-g+a T-g+1+a
T-+a=T-g+a
T+aT+a
The if I do the same to things but with this time 'b' I will get
T-17 T-18 T-1T-1+b T-18+b T-17+bT-g-1-gb+a T-g+a T-g+1+gb+a
T-=T++b=T-+gb+a
TT+bT+gb+a
So now if I put all this into formulas
T+gb+a+t-g+gb+a+t-g-1+gb+a+t-g+gb+a+t-g+1+gb+a=
5t(t+a)+g(-7+5b)
0=5(t+a)+g(-7+5b)
180=5(t+a)+g(7+5b)
0=5(t+a)+g(-7+5b)+7(g+1)
70=5(t+a)+g(-7+5b)+7(g-1)
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