So far the formulas that I have found are;
9x9 Grid; T=5n-63
8x8 Grid; T=5n-56
7x7 Grid; T=5n-49
6x6 Grid; T=5n-42
5x5 Grid: T=5n-35
Now that I have found the individual formulas I shall check that they are correct using a different T- number of the particular grid in question.
9x9 Grid;
T-number used= 78. T=5n-63. So 78x5=390-63=327. The t total I have found is this. I shall test it manually now for this grid size and the four other ones.
Test= 78+69+60+59+61=327. This formula works.
8x8 Grid;
N=63 63x5=315-56=259
Test- 63+55+46+47+48=259. This formula works.
7x7 Grid;
N=47 47x5=235-49=186
Test- 47+40+32+33+34=186. This formula works.
6x6 Grid;
N=35 35x5=175-42=133
Test- 35+29+22+23+24=133. This formula works.
5x5 Grid;
N=24 24x5=120-35=85
Test- 24+19+13+15+15=85. This works. Now that all my formulas work I shall move on to the general formula.
The General Formula
After finding the formula for five different grid sizes it will now be necessary to find the general formula so that I can work out the T-total for a grid of any size and not just the grid sizes I have already done.
I will need to use the formula that I have already found to help me;
9x9 Grid; 5n-63
8x8 Grid;5n-56
7x7 Grid;5n-49
6x6 Grid;5n-42
5x5 Grid;5n-35
To help me find the formula I shall put the formulas in a line and find the different.
63 56 49 42 35
-7 -7 -7 -7
The difference is constant so the general formula will contain a seven in it. The general formula will also contain a five as there is one in all of the separate formulas.
I will use ‘g’ to represent the grid size from now on.
The final formula for the T-total on any grid size is therefore 5n-7g
So, for a 9x9 Grid size the formula would be:
5n- where n is the t-number.
7g= 7x9- this is from 9x9
so 5n x(7x9)= 5nx56
This is the general formula for any grid size to find the T-total.
Testing the General formula On A Different grid Size.
Now that I have found my formula and found that it works for a grid size that I have chosen I need to prove that it works for a grid size that I have not just tried and experimented with. I have decided to test my formula on a 10x10 grid:
My formula- T=5n-7g
I predict that if my T-number is 63 then my T-total will be 245. I will prove this below on a 10x10 Grid.
42+43+44+53+63=245.
This total is the same number that I found with my general formula above. This is the test that proves that my General formula works for any size of grid and not just the ones that I tested and experimented with previously.
Why is there a 5 and a 7 in the Formula?
The Five:
Using a 9x9 grid for an example I shall prove why there should be a 5 in the formula.
Looking at the grid you can see that there are 5 separate lots of n in the t shape. Looking further at the pattern we find this sum to work out the total sum for the grid.
n+(n-9)+(n-19)+(n-18)+(n-17). This sum equates to this; 5n-63. There are five lots of n so this is where we get the 5 in the formula from. Also looking at the pattern for a 9x9 grid we can see where the 5 comes from.
T= 20 21 22 23 24
N=37 42 47 52 57
+5 +5 +5 +5
There being a difference of +5 in the sequence means that the formula will have a 5 in it somewhere. This ‘5’ ends up being the 5 and is in the general formula for the T-total.
The 7:
Using gxg to represent any grid size we can prove where the 7 comes from by doing the following.
Converting this into an actual t-shape off a real grid will give us the following.
For this example I will use a 9x9 grid.
Using this example we can work out what the total for this equation will be. It will be as follows:
(n-2g-1)+(n-2g)+(n-2g+1)+(n-g)+(n). This is equal to 5n-7g. To get this I added all the negative ‘g’s’ and that gave me a total of -7. This is how I got to my seven in the formula, this proves that there must be a seven in the formula for it to work correctly. You can also see that there are five lots of ‘n’ in this formula again. This five is from the general formula for the t-total. This is why there must be a five and a 7 in the general formula.
Translating the T and the effect it has on the T-total.
Now that I have found the general formula for the t-total on any grid size I must take the investigation one step further. I must now investigate the effect of translating the shape to a different point on the grid.
To do this I will use three different grid sizes, 9x9, 8x8 and finally 7x7. This will enable me to find a range of different formula for me to compare. After finding the different formula I will find the general formula for any grid size. The translations will be done using vectors. I will use this to represent the vectors in my work.
The top will be (x)
And the bottom (y)
Vector Tables for 9x9, 8x8 and 7x7 Grid Sizes.
I will now find the t-total of the new shape and the change in the t-total from the original grid size to the new grid size. I will set out my results in a table so that they are clear and easy to read from.
9x9 Grid. My T-number for this Grid will be 50 to start with.
After looking at this table I have found the Formula for the 9x9 grid transformations. It will have to contain the original general formula that I found previously but with an extra part added onto the end. The formula looks like this T=5n-63+5x-45y
I will now have to repeat this for an 8x8 grid size so that I have more than one example to go off to help me find the general formula.
8x8 Grid Size: My T- number for this grid size will be 36 to start with.
Looking at this formula again I can see that the formula for an 8x8 grid using the vectors above is T=5n-56+5x-40y.
I will now repeat this a third and final time on a 7x7 grid so that I have three different sets of results to work with and to find the general formula from.
7x7 Grid Size: My T- number for this grid size will be 32 to start with.
This is the final table for vectors that I will need and the final formula that I will need to find the general formula. The formula for this final 7x7 grid is T=5n-49+5x-35y.
General formula for transformations on any grid size
I will now need to find a pattern in the three formulas that I have found to help me find the general formula.
T=5n-63+5x-45y- 9x9 Grid
T=5n-56+5x-40y- 8x8 Grid
T=5n-49+5x-35y- 7x7 Grid
The formulas all contain a multiple of their grid size in which when multiplied by 7 gives the second number in each formula. They all also contain the 5n from the original formula so this too must be included. Also each of the formula contain a 5x so this will be in the general formula. There must also be a pattern in the numbers at the end of the specific formulas to help me find the general. Each time the number on the end goes down by five. For the mean while I write out the formula that I have so far to help.
T=5n-7g+5x..
I will test this on a 9x9 grid to see what the total is so far. I will use a T-number of 50 and a vector of (0,1)
250-63+0=187
the number I should be getting is 45 less than this.
The general formula for any grid size translation is T=5n-7g+5x-5gy.
This will work on any grid size.
Testing the Formula
It is important that I test the formula that I have found using different vectors so that I can check I have found the correct formulas that work for anything and that it is not just a coincidence that it has worked for me so far. I will test my formula on a 9x9 grid.
I will choose the vectors(1,2) for this test. The original shape will be green and the new shape red. I will use this formula T=5n-7g+5x-5gy.
I predict that the t-total of the new shape will be 55*5-70+5-100=110. I will add up the numbers off the new shape on the grid to check whether the formula really does
Work.
15+16+17+26+36=110. this is the same answer that I got using my general formula so it has been proven to be correct.
Rotating the T-shape by 90 degrees, -90 degrees and 180 degrees.
I am now into the third part of the investigation and it will be necessary to investigate the effect that rotating the T- shape has on the T-total and the numbers that are in the shape. To do this I will rotate the T-shape by set denominations on two separate grid sizes to give an accurate picture to find the formula that is required.
I will rotate the t-shape by 90 degrees, -90 degrees and 180 degrees
, by doing this for three different grids, 9x9, 8x8 and 7x7 it should enable me to build up an accurate picture of any patterns that develop. I do not however expect that I will find just one general formula for a rotation of any degrees, there should be three separate formulas for 90 degrees, -90 degrees and finally 180 degrees.
This third part of the investigation will help immensely in the final part of the investigation which will be a rotation and a translation at the same time.
Finding The formula rotations on a 9x9 Grid
The table below will be used to translate the original t shape with a T-number of 50. I will now rotate the t-shape three time on the grid below.
The original total of the T-shape is;
31+32+33+41+50=187
The +90 degree rotation is; 50+51+52+43+61=257
The -90 degree rotation is: 39+48+87+49+50=243
The 180 degree rotation is; 50+59+67+68+69=313
These are the three totals of the new t-shapes. Looking at the three sums above and using the 5n from before to help there is a pattern:
For +90 degrees the formula I have found is 5n+7
For -90 degrees the formula I have found is 5n-7
For +180 degrees the formula I have found is 5n+7g
These three formulas work perfectly on this grid size but I shall have to do another grid size to be sure that they are correct.
Rotation on a 8x8 Grid Size.
The t-number that I will use this time will be 36.
The original total of the t-shape is; 19+20+21+28+36= 124
The +90 degree rotation is;
36+37+30+38+46=187
The -90 degree rotation is;
36+35+34+26+42=173
The 180 degree rotation is; 36+44+51+52+53=236. Using these three new totals I will test my formula.
+90 degrees= 5n+7, so 5x36+7=187. This formula works.
-90 degrees= 5n-7, so 5x36-7=173. This formula works
The 180 degree rotation= 5n+7g, so 5x36+(7x8)=236. This formula also works so I have tested all my formula and found that they are all correct.
So the general formula for the three separate rotations is as follows:
+90 degrees; 5n+7
-90 degrees; 5n-7
180 degrees; 5n+7g
These can be used to work out the rotations on any grid size.
Part Four; A Rotation and a Translation.
This is the final part of the investigation; I will now need to find out what the relationship between the rotations and translations on a specified grid size are. This will hopefully help me find three separate final general formulas that can be used to work out the t-total of any new shape that have been transformed, rotated or even both. I will able to use the formula that I have already found in previous stages of the investigation and should just be able to piece little bits together to get the formula.
I will need to analyse all the information that I found in the rotations to help me. The reason that I will end up with three separate formulas is because there are three separate formulas for rotations. I cannot group them all together in one because it would not give me the correct answer so they must be left separate in the final general formulas.
180 degree general formula for Rotation and Translation
I am now going to find the general formula that will work for a 180 degree rotation from a point outside the T- shape. There will be three separate formulas for +90 degrees, -90 degrees and 180 degrees. I will use a 9x9 grid for this part of the investigation.
The shape on the left is the original and the new shape is on the right.
22 becomes 60- difference of 38
24 becomes 61- difference of 37
23 becomes 59- difference of 36
32 becomes 50- difference of 18
The t-number stays the same throughout.
The differences now need to be found to find the formula.
Differences= 0 equal to 0
18 2g
34 36 38 4g-2 4g 4g+2
2g+4g-2+4g+4g+2=14g. This will be in the formula for 180 degree rotation and translation.
Using the vectors from the tables in the previous pages we can find the final formula. Looking at what the formula is for a translation on a 9x9 grid I should be able to find the formula.
T=5n-7g+5x-5gy. If I add the 14g to this then I should have the final formula for a 180 degree rotation. The final formula is 5n-7g+5x-5gy+14g, this will simplify to;
5n+7g+5x-5gy. I must now repeat this for -90 and +90 degrees. Then I will prove that my formulas are correct.
+90 degree general formula for Rotation and Translation
I will now have to repeat what I did above but for +90 degrees. How I will do this is below.
Using a 9x9 grid I will once again find the differences. The one on the left is the original and the one on the right is the one that is the final shape once it has been rotated.
24 becomes 52-difference of 28
23 becomes 43-difference of 20
22 becomes 34- difference of 12
32 becomes 42- difference of 10
T number (41) stays the same
Difference:
12 g+3
0 10 20 0 g+1 2g+2
28 3g+1
If we add all this up, 0+(g+1)+(g+3)+(2g+2)+(3g+1), then it equals 7g+7
Using the vectors from the tables in the previous pages we can find the final formula. Looking at what the formula is for a translation on a 9x9 grid I should be able to find the formula.
T=5n-7g+5x-5gy. If I add the 7g+7 to this then I should have the final formula for a +90 degree rotation. The final formula is 5n-7g+5x-5gy+7g+7, this will simplify to;
5n+5x-5gy+7. I must now repeat this for -90 degrees. Then I will prove that my formulas are correct.
-90 degrees general formula for Rotation and Translation
I will now find the last formula that I will need. This will be for -90 degrees. I shall use the same procedure as I have done when I found the previous two general formulas. I will use a 9x9 grid, I will show my working out below.
22 becomes 48- difference of 26
23 becomes 39- difference of 16
24 becomes 30- difference of 6
32 becomes 40- difference of 8
The t-number (41) stays the same.
Difference:
26 3g-1
0 8 16 0 g 2g-2
6 g-3
If we add up all the differences g+(3g-1)+(2g-2)+(g-3), then we get 7g-7.
Using the vectors from the tables in the previous pages we can find the final formula. Looking at what the formula is for a translation on a 9x9 grid I should be able to find the formula.
T=5n-7g+5x-5gy. If I add the 7g-7 to this then I should have the final formula for a -90 degree rotation. The final formula is 5n-7g+5x-5gy+7g-7, this will simplify to;
5n+5x-5gy-7.This is how I have found all of my formulas for the rotations and the translations.
So the final formulas are:
For 180 degree rotation; 5n+7g+5x-5gy
For +90 degree rotation; 5n+5x-5gy+7
For -90 degree rotation; 5n+5x-5gy-7
