If we use all the information of each section above together we can get a general formula for a grid of any size and any T-shape that we wish.
t = 5x – 7g
Translations
If we move the T-shape 1 Square to the right the T-total leads to an increase of +5. A translation of 1 square to the left leads to a decrease of –5. And if we translate the T-shape upwards 1 square this leads to a decrease of –45 if it is translated down it leads to an increase of +45. If we start with a Vertical translation we can work out a general formula for any size grid.
(Vertically)
I started to look at a 9 by 9 grid.
If we start with the 1st T-shape drawn on the grid we get a T-number of 50 and a T-total of 187. If we than translate the T-shape up to have the T-number as 41 and then the T-total is 142. We see that there is a difference of 45.
If we keep moving the T-shape up 1 space at a time we can collect data and put it in a table.
As we can see there is a trend of 45 on a 9 by 9 grid if the translation is +1. I then looked at grids of different sizes.
6 X 6 grid
In this there is a common trend of 30. Both 45 and 30 are multiples of 5. I worked out that if you translate the T-shape up one square you time’s the grid size (g) by 5
For a movement upwards you use this formula (-5g) where g is the grid size
To get the number you take away, for example, on a 10 by 10 grid we can predict that if you make a translation of +1 you will have to take away 50 from the original T total.
If we take a T-shape that has a T-number of 79 we can work out the T-total by the formula
t = 5x – 7g
t = (5 x 79) – (7 x 10)
t = 395 – 70
t = 325
If we move the translation up one square you should get the result of:
t = 5x – 7g
t = (5 x 79) – (7 x 10)
t = 395 – 70.
t = 325 – 5g
t = 325 – 50
t = 275.
We can now check this by using the original formula filling it in with the data for a T-shape with a T-number of 69.
t = 5x – 7g
t = (5 x 69) – ( 7 x 10)
t = 345 – 70
t = 275
This proves that you have to use -5g for an upward translation of 1. If you wanted to do a translation of up3 you would then have a formula of p(–5g) where p= number of translations. So if you wanted to do a translation of up3 you would have to put the 3 in where the p is and then work out the rest of the formula.
The full formula looks like this.
t = (5x – 7g) – p(5g) p = number of translations
x = T-number
g = grid size
t = T-total
Translations (Horizontal)
We can then use the same method to work out the formula for a translation horizontally. If we start with a grid of 9 by 9.
If we start with the T-number of 39 this gives us a T-total of
t = 5x – 7g
t = (5 x 39) – (7 x 9)
t = 195 – 63
t= 132
And then translate it horizontally by 1 to the right this gives us a T-number of 40 and a T-total of
t = 5x – 7g
t = (5 x 40) – (7 x 9)
t = 200 – 63
t= 137
As we can see there is a difference of +5 as the T-shape moves horizontally right 1 square. If we then do more translations of +1 right from the T-shape with a T-number of 40, and put all the results into a table you can get an idea of where to start a formula.
So if we want to write a formula for a translation of a T-shape on a five by five grid right +1 or +p we get the formula of:
t = 5x – 7g + (p5) p = number of translations
x = T-number
g = grid size
t = T-total
We now need to look at grids of different sizes to get a universal equation for a translation on the horizontal axis.
10 by 10
If we put all the translations with a T numbers between 73 and 79 in a table, we can compare the results to them of a 9 by 9 grid.
As we see the difference is still +5 so the formula for a translation for any horizontal translation is:
t = 5x – 7g + (p5) p = number of translations
x = T-number
g = grid size
t = T-total
Combinations of translations (diagonal)
For diagonal translation across a grid a combination of horizontal and vertical translations are used, therefore I predict that if I combine my 2 found equations for horizontal and vertical movement’s, I can generate a general formula for diagonal translations. We need now to combine the two equations. I only need to instance of 5x – 7g as only one T-shape is being translated:
t = (5x – 7g) – b(5g) + (a5)
a = number of translations horizontally
b = number of translations vertically
x = T-number
g = grid size
t = T-total
To prove this equation we need to again start with our standard grid and position and try it on a combination translation.
The above translation is a combination of a horizontal movement of +3 and a vertical translation of +3 also. The original T-shape has a T-total of 217, and the translated T-shape has a T-total of 97, using the equation we will try and generate the second T-total to prove our theory correct.
t = (5x – 7g) – b(5g) + (a5)
t = ( (5 x 56) – (7x9)) – 3 x( 5 x 9) + (3 x 5)
t = (280– 63) – 135 + 15
My equation has been proved correct using this translation, we must now try it on another grid size with another type of a combination translation, to verify that it is correct; I have chosen a grid size of 7 by 7.
Here we can see a translation of +4 vertically and +4 horizontally. The original T-shape has a T-total of 171, and the translated T-shape has a T-total of 51. Using our generated formula we can see if it is correct for this grid size, or do we have to modify the formula to relate to the grid size.
This proves my equation correct as the correct translated T-total is generated.
t = (5x – 7g) – b(5g) + (a5)
Only if you split the diagonal translation into two parts the horizontal movement and a vertical movement you can use this formula to generate you new T total
It is also apparent that this equation can be used for any singular vertical or horizontal translation, as the other value (vertical or horizontal) will be equal to 0, therefore cancelling out that part of the equation, as it would equal 0.
Any type of translation (vertical, horizontal or a combination) can be found by using the equation of
t = (5x – 7g) – b(5g) + (a5)
Rotations
To begin with I shall try to find generalisations and rules for rotations of T-shapes, where the T number is the centre of rotation. Two other variables also need to be defined: the amount to rotate by e.g. 90, 180 or 270 degrees and the direction in which to rotate e.g. clockwise or anti-clockwise. To begin with, I will use a basic 9 x 9 grid.
Firstly, we can see our T-shape with a T-total (t) of 142 (yellow), then a rotation of the T-shape 90 degrees clockwise around the point 41 (orange). The T-total of this shape is now 212. If we rotate our T-shape by 180 and 270 degrees clockwise, again it will be easier for us to build up a profile and some generalisations.
If we put all the results from the 9 by 9 grid into a table, this will allow us to look at the results more clearly and try to make a generalisation. To help us start to form a formula
It is hard to make any immediate generalisation at this point. If we try the same rotations on a different grid width and plot the results in a table, patterns might become easier to see. I have chosen a grid width of 6 by 6 to try this on. We will then plot all the results from every table together.
There is no obvious equation to be seen from these two sets of results. If we take each part as an individual equation it might become apparent on what to look at.
Looking at a rotation of 180o / A flip
Firstly a rotation of 180o / A flip will be easy to find, as it is a 180 degree rotation of the original T shape is the exact opposite to the original so the equation will be plus a multiple of some number.
Working this out using the traditional method the answer is 31 (2 + 5 + 7 + 8 + 9), using the formula
t = 5x – 7g
t = (5 x 2) - (7 x 3)
t = 10 - 21
t = -11
As we can see that the formula does mot work if we leave the T number as the original positions, located at the bottom of the shape. The original formula is no good. To double check that we have to use a different formula we can try it on a different grid size.
5 by 5
Again using the traditional method, we get 75 (8 + 13 + 17 + 18 + 19), using the formula:
t = 5x - 7g
t = (5 x 8) - (7 X 5)
t = 40 - 35
t = 5
5x - 7g cannot be used to find the T-total (t) of a rotation of 180o shape, on any grid size, but I then thought that if the formula for an original t shape is t = 5x – 7g if it is rotated of flipped to 180o the formula should flip to. So I think that the formula should now read this
t = 5x + 7g
But at the moment we do not know if this is right so we have to look at it on some grids. First I will look at the original 9 by 9 grid.
For the original T shape that is in the colour of yellow we know for a fact that the formula is
t = 5x – 7g. But we are going to look at the rotated T shape of 180o this has a T total of 268. If we keep the t number of 41 and fill it into the new predicted formula of t = 5x + 7g. If we substitute in the vales hopefully it will work.
t = 5x + 7g
t = (5 x 41) + (7 x 9)
t = 205 + 63
t = 268
As we can see my prediction is correct. If you have a rotation of 180o the formula has the same begging of t = 5x and only the ending changes to +7g instead of – 7g
Moving on to rotations of 90 and 260 degrees
As we know that a 180-degree flip will change to become a negative equation, we only need to work out one kind of shape and then be can reverse the formula to get the other. Therefore we shall work out formulas for a rotation of 90 degrees clockwise.
Again using the old method we get a T-total of the rotated shape is 92. We have to start from scratch to make a formula so we can follow the steps used to find the original formula for the T shape.
t = (17 + 0) + (17 + 1) + (17 + 2) + (17 – 3) + (17 + 7)
So if we substitute 17 with r we get:
t = r + (r + 1) + (r + 2) + (r + 1) + (r + 4)
t = 5r + 8
t = 5r + 8
On any grid the formula 5r + 8 can be used to work out the T-total (t) where r is the T number. The formula t = 5r + 8 can be used to find the T-total (t) of any T shape that has been rotated 90 degrees clockwise where r is the T number.
Following the 180 degree flip rule we can predict that for T shapes which has been rotated by 270 degrees the formula is t = 5r – 8. As it is a flip of the T shape with a rotation of 90 degrees clockwise.
Working this out by hand, we get a T-total of 63 (7 + 12 + 13 + 14 + 17) for this T-shape. If we use the formula, we get:
t = (5 x 14) + 8
t = 70 + 8
t = 78
The formula t = 5x + 8 can be used to find the T-total (t) of any shape rotated 90 degrees clock wise on any sized grid, where x is the T number. The formula t = 5x – 7 can be used to find the T-total (t) of any shape rotated 270 degrees clock wise on any sized grid, where x is the T-number. Therefore we can state from a standard T-shape position the following equations can be used to generate the T-total from x.
These are all of the formulas used to find any rotation.
I have also rotated and flipped the formulas so that we can have formulas for an anti clockwise rotation.