275 – 70 = 205
I multiplied it by 55 as it is the T number and according to the formula, the T number is meant to be multiplied by 5.
These tests show that my formula for working out the T total in an upright T shape in a 10 by 10 grid is correct.
I will now do the same method to work out the formula for working out the T total in an upright T shape in a 12 by 12 grid
12 by 12 Grid
1) 1+2+3+1+26 = 46 T = 46 X = 26
After observing different upright T shapes in a 12 by 12 grid I found that the format was this:
This was the format because the number above X was usually 12 taken away from it and the number above that was once again double the number below. On each side of the X – 24, the number was either 25 taken away from X and 26 taken away from X.
I will add the numbers together to give me a T total:
X + X – 12 + X – 24 + X – 25 + X – 26 = T
This becomes 5X – 84 = T
I will test these to confirm that the formula is correct:
CHECK:
I will first try it on the example I first used: 5 x 26 = 130
130 – 84 = 46.
So far my formula for working out the T total in a T shape that is upright in a 12 by 12 grid I will test it once more.
CHECK:
5 x 65 = 325 40+41+42+53+65 = 241
325 – 84 = 241
These show that my prediction for the formula is correct.
After trying different grid sizes to work out a formula for the T total I came up with the following formulas:
5X-63 = T (for a 9 by 9 grid.)
5X-70 = T (for a 10 by 10 grid.)
5X - 84 = T (for a 12 by 12 grid)
From these, I have to find a formula that will enable me to work out the T total in an upright T shape no matter what the grid size.
Firstly, I realised that in each formula the number after the minus sign was a multiple of 7. This gave me the idea that that the 7 would be in the formula.
Using this I saw that when you multiply 7 by 9 in a 9 by 9 grid you get 63.
In a 10 by 10 grid, I had to multiply 7 by 10 to get to 70 and in a 12 by 12 grid I had to multiply 7 by 12 to get to 84.
From looking at these steps I realised that 7 has to be multiplied by the size of the grid in order to get the formula for the T total in an upright T shape. Using this evidence, I came up with the following formula:
T = 5X – 7L
L= length of the grid.
I will test this formula to confirm that it is the correct formula for working out the T total in an upright T shape in any grid. I will check it in an 11 by 11 grid and a 7 by 7 grid.
CHECK:
- 1+2+3+13+24 = 43
5 x 24 = 120
11 x 7 = 77
120 – 77 = 43
- 1+2+3+9+16 = 31
5 x 16 = 49
7 x 7 = 49
80 – 49 = 31
These tests show that my formula for working out the T total in an upright T shape in any size grid is correct.
I have also come up with the following T shape format for an upright T shape that is in any size grid:
I thought that this was the format because form looking back at the T shapes in the other grids I found that I had to minus the length from X to give me the number above the T number. Then for the number above that I had multiply the length by two and take it away from X (T Number). Then on each side of that number you had to add or minus 1 from X – 2L. This gave me the upright format as seen above, but to confirm it I will test it as usual:
I will add them as I did before:
X + X – L + X – 2L + X – 2L – 1 + X – 2L – 1 = T
This becomes 5X – 7L = T
My formula for working out the T total in an upright T shape in any grid size is correct.
I will now rotate the T shape to see how the format of the T shape changes and how the formula for working out the T total will change.
90 DEGREES CLOCKWISE:
After turning the T shape 90 degrees clockwise I found that this was the general format for a T shape that has been rotated 90 degrees clockwise.
I thought that this was the format because the two numbers following on from X were consecutive and so were obviously X + 1 and X + 2. The number on top of X + 2 I found was X + 2 but L (length of grid) also taken away from it. The number below X + 2 was X + 2 but L added to it.
I will add these as I have done before to give me my T total:
X + X + 1 + X + 2 + X + 2 – L + X + 2 + L = T
This gives me a formula of T = 5X + 7.
This formula that will work out the T total in a T shape that has been rotated 90 degrees clockwise. I will test this formula to check it is correct. I will use a 6 by 6 grid to check it on.
CHECK:
9+10+11+4+150= 52
9 x 5 = 45
45 + 7 = 52
This not only confirms my formula but my general format of a T shape that has been rotated 90 degrees clockwise.
I will now turn it 180 degrees clockwise to see the difference in the format of the T shape and the formula for solving the T total if there is any.
180 DEGREES CLOCKWISE
By turning the T shape 180 degrees clockwise I noticed that the T shape format did change and this was what it became:
I found that the number below X was a number that had the length added to the T number. Furthermore the number below X + L was doubled to become X + 2L. To the left of this number it was X + 2L - 1 as it was one less and the number on the right was one more than it making that number X + L + 1. Not only, this but it is symmetrical to the upright format.
I will add these terms together as usual to give me a T total:
X + X + L + X + 2L + X + 2L + 1 + X + 2L – 1 = T
This will simplify to T = 5X + 7L.
This has given me a formula to work out the T total in a T shape that has been rotated 180 degrees clockwise. I will check it works in a 6 by 6 grid:
CHECK:
9+15+20+21+22 = 87
5 x 9 = 45
7 x 6 = 42
42 + 45 = 87
This shows that my formula for working out the T total in a T shape that has been rotated 180 degrees clockwise is correct.
I will now rotate the T shape 270 degrees clockwise.
270 DEGREES CLOCKWISE
I have found that that the general T shape format after being rotated 270 degrees clockwise is:
I thought that the format was this because the numbers to the right are consecutive to X and so they would decrease in value hence X – 1 and then X – 2. Above X – 2 I found that you had to minus the length from it to get to that and below X – 2 you had to add the length to give you that number. This format is also symmetrical to the format of the 90 degrees T shape.
I will add the terms in the T shape to give me my T Total:
X + X – 1 + X – 2 + X – 2 + L + X – 2 – L = T
This turns into T = 5X – 7.
As I did before I will test this formula on a 6 by 6 grid.
CHECK:
3+9+10+11+15 = 48
5 x 11 = 55
55 – 7 = 48
This confirms my formula for working out the T total in a T shape that has been rotated 270 degrees clockwise.
From turning the T shape three times I have come up with the following T shape formats and T total formulas:
T = 5X – 7L T = 5X + 7 T = 5X + 7L T = 5X – 7
I will now add a vector to see what effect it has on the T shape formats and the T total formulas.
I found that after adding a vector the following formula helped you to get from your original T number to your new number:
X + A – BL = X (NEW T NUMBER)
X = T NUMBER
A = MOVEMENT IN X AXIS
B = MOVEMENT IN Y AXIS
L = LENGTH
I will start off by applying a vector to the upright T shape. The format of the upright T shape is effected by the vector and becomes:
I came to this T shape because you have moved the X across the X axis it will have a new value and that will be the value of the vector for the x axis, but then as you apply the second part of the vector you have to take away the movement in the y axis multiplied by the length of the grid to come to your new T number. Whatever occurs to the T number happens to the rest of the T shape also and that is why they all have the same endings.
Just as before I will add the terms within the T shape to give me the T total for an upright T shape once a vector has been applied:
X + A + BL + X – L + A – BL + X – 2L + A – BL + X – 2L – 1 + A – BL + X – 2L + 1 + A – BL = T
This simplifies to T = 5X + 5A – 5BL – 7
This is the formula for working out the T total of a T shape that has had a vector applied to it.
I will test it on a T shape in a 10 by 10 grid and use the vector (5/6):
The first T number will be 62; by moving it across 5 it will become 67. Then by moving it up 6 the T number will become 7. I will now apply the formula for finding the new T number.
62 + 5(A) = 67
6(B) x 10 (L) = 60
67 – 60 = 7
This has given me the same result as before this shows that my formula is correct and so is the T shape.
I will now apply a vector to a T shape that has been rotated 90 degrees clockwise.
90 DEGREES
The T shape after being rotated 90 degrees clockwise changes once a vector had been applied to it this is the new format.
This is the format because once again the formula for getting to the new T number is added on to the T number and if it is added to the T number then it has to be added to the rest of the T shape as the other terms move in the same way.
I will add these terms to give me a formula for working out the T total of a T shape that has been rotated 90 degrees clockwise and had a vector applied to it:
X + A - BL + X + 1 + A – BL + X + 2 + A – BL + X + 2 – L + A – BL + X + 2 + L + A + BL = T
This becomes T = 5X + 5A + 5BL + 7
This is the formula for working out the T total in a T shape that has been rotated 90 degrees clockwise and had a vector applied to it.
I am going to move on to applying a vector to on a T shape that has been rotated 180 degrees.
180 DEGREES
Once a vector has been applied to a T shape that has already been rotated 180 degrees it becomes:
This is the format because once again the formula for getting to the new T number is added on to the T number and if it is added to the T number then it has to be added to the rest of the T shape as the other terms move in the same way.
I will add the terms in the T shape to provide me with a T total for a T shape that has been rotated 180 degrees clockwise and had a vector applied to it:
X + A – BL + X + L + A – BL + X + 2L + A – BL + X + 2L + 1 + A – BL + X + 2L – 1 + A – BL = T
This can be simplified into T = 5X + 5A – 5BL + 7L
This is the formula for solving the T total in a T shape that has been rotated 180 degrees and had a vector applied to it.
I am now trying a vector on the T shape that has been rotated 270 degrees clockwise.
270 DEGREES
I have come up with the following format for a T shape that has been rotated 270 degrees clockwise and had a vector applied to it:
I came to this T shape because as you have moved the X across the X axis it will have a new value and that will be the value of the vector for the x axis, but then as you apply the second part of the vector you have to take away the movement in the y axis multiplied by the length of the grid to come to your new T number. Whatever occurs to the T number happens to the rest of the T shape also and that is why they all have the same endings.
I will add the terms in the new T shape format to give me a T Total:
X + A – BL + X – 1 + A – BL + X – 2 + A – BL + X – 2 – L + A – BL + X – 2 + L + A – BL = T
This turns into T = 5X + 5A – 5BL – 7.
From applying vectors to the four different T shapes I have come up with the following formulas along with the formulas without vectors applied to them.
After looking at these formulas, I found that once a vector had been applied they change. In the new formulas 5A – 5BL has been added to them. This is because the formula for applying a vector is X + A – BL. As there are 5 terms in the T shape the formula has to be applied 5 times. This brings me to the formulas for the T total once having added a vector. So, basically the formula stays the same but has 5A – 5BL added to it.
