However this formula is only correct for G8 and N can only be placed in the places not shaded in on the second grid.
Now I will find the overall formula for G7, however I will only use the algebraic approach:
This table explains the t-shape diagram
To find T = mN + c I must add up what is in the t-shape and simplify the out come:
T = N + N- 7 + N- 15 + N- 14 + N- 13
T = 5N – 49
Therefore the overall equation is T = 5N – 49
To test this I have randomly put a t-shape into grid 1.
N = 26
T = (26 + 19 + 11 + 12 + 13) = 81
Therefore: 81 = (5 × 26) – 49 = 81
I will now test this formula again; this is because I have only used the algebraic approach. Therefore I want to double check
N = 46
T = (46 + 39 + 31 + 32 + 33) = 181
Therefore: 181 = (5 × 46) – 46 = 181
Therefore the formula of T = 5N – 49 is correct
However this formula is only correct for G7 and N can only be placed in the places not shaded in on the second grid.
Summary
To summarise I will collate the formulas that I have concluded and insert them into a table. Then I will use a random grid size and test the formula I conclude.
However I will only use G3 to G10, the reason that I cannot go below G3 is because the t-shape cannot fully fit the grid. I will also not choose one of G that I have already used, this is because if I do use the grids that I have already done then I cannot accurately prove that my formula is correct.
I have noticed that in all of the formulas it is consistent that 5N is in the formula. Also the second term in the formula is the sum of G × -7.
Therefore I predict that the general formula is T = 5N – 7G
I will now test my prediction by using G10 and grid 11. I will use N = 25
T = (5 × 25) – (7 × 10)
T = 125 – 70
T = 55
Now I will use addition to see if T is the same as when I used the formula
T = 25 + 15 + 4 + 5 + 6 = 55
I can now say that the forula that I predicted is correct, this is because when I used the predicted formula the answer I got to was 55 and when I used addition the answer I got was the same of 55.
However if you look at G10 in grid 11 when N = 25 there is a relationship between T, N and G. this is:
I will now add up all of that is in the t-shape and put it into its simplest form:
T = N + (N – G) + (N – 2G) + (N – 2G + 1) + (N – 2G – 1)
T = 5N – 7G
Therefore this also correlates with the formula that I previously found, therefore the formula of T = 5N – 7G is correct.
Part 3
Here I will investigate the effect of a translation (x/y) on t-total.
Whilst doing this investigation I will use T2 as the new t-total.
Horizontal translation (x/0) for all grid sizes:
Firstly I will use G8 to find the effect of (1/0).
I can say that the formula for the t-shape in grid 3 is 5N – 7G, this is because I proved it in part 2. Now to find the formula in grid 4 compared to 5N – 7G
T in grid 3 = 34
T in grid 4 = 39
Here it shows that (1/0) is 5 more than (0/0). Therefore the formula here is
T2 = 5N – 7G + 5
This will work for any G; this is because as the t-shape moves along it increases by 1. Therefore as there are 5 squares in each t-shape, the value moves up in 5’s
However now I have found that for every time that N is moved horizontally once, T increases by 5. Therefore I predict that for (2/0) T will increase by 10 for any grid size; so I have chosen a grid size at random, which happens to be G6, to test my prediction on. The values for N I have also chosen to use are 14 and 16, where the t-shape for N14 is in grid one and the t-shape for N16 is in grid two.
T in grid 1 = 28
T in grid 2 = 38
Here it shows that (2/0) is 10 more than (0/0). Therefore the formula here is
T2 = 5N – 7G + 10
This shows that my prediction is correct of that for every time N moves horizontally once T increases by 5.
Therefore I can make an overall prediction for T2. This is:
T2 = 5N – 7G + 5x (here x is the number of places moved horizontally by N)
To test that the general formula for the horizontal translation of the t-shape, I will again choose 2 G’s at random, and test this twice; so that I can make an accurate conclusion.
Firstly I have chosen G7, where the values for N I have also chosen to use are 23 and 27, where the t-shape for N23 is in grid three and the t-shape for N27 is in grid four; also I have used a translation of (4/0). Therefore I expect the formula to be
T2 = 5N – 7G + (5 × 4):
T in grid 3 = 66
T in grid 4 = 86
Therefore the formula here is:
T2 = 5N – 7G + 20
This means that I was correct in my prediction.
Secondly I have chosen G10, where the values for N I have also chosen to use are 32 and 39, where the t-shape for N32 is in grid three and the t-shape for N39 is in grid four; also I have used a translation of (7/0). Therefore I expect the formula to be
T2 = 5N – 7G + (5 × 7):
T in grid 3 = 90
T in grid 4 = 125
Therefore the formula here is:
T2 = 5N – 7G + 35
This again means that I was correct in my prediction.
Therefore overall this means that the general formula for the horizontal translation of the t-shape (x/0) is: T2 = 5N – 7G + 5x. Also here N is the t-number of the original t-shape.
Now I will find the general formula for the vertical translation (0/y).
I will firstly use G8 to find translations of (0/1), (0/2) and (0/y). The values for N I have also chosen to use are 58 and 50, where the t-shape for N58 is in grid five and the t-shape for N50 is in grid six.
T in grid 5 = 234
T in grid 6 = 194
Here it shows that (0/1) is 40 less than (0/0). Therefore the formula here is
T2 = 5N – 7G – 40
The values for N I have also chosen to use are 62 and 46, where the t-shape for N62 is in grid five and the t-shape for N46 is in grid six.
T in grid 5 = 254
T in grid 6 = 174
Here it shows that (0/2) is 80 less than (0/0). Therefore the formula here is
T2 = 5N – 7G – 80
From these I can see that there is a trend connecting G, y and 5. This is (5 × G × y).
Therefore from this I can make a prediction for the general formula for any grid size and any vertical translation. My prediction is T2 = 5N – 7G – (5 × G × y) I will now test this on a random grid twice.
Firstly I have chosen G9, where the values for N I have also chosen to use are 80 and 44, where the t-shape for N80 is in grid three and the t-shape for N44 is in grid four; also I have used a translation of (0/4). Therefore I expect the formula to be
T2 = 5N – 7G - (5 × 9 × 4):
T in grid 3 = 337
T in grid 4 = 157
Therefore the formula here is:
T2 = 5N – 7G - 180
This means that I was correct in my prediction.
Secondly I have chosen G6, where the values for N I have also chosen to use are 32 and 20, where the t-shape for N32 is in grid three and the t-shape for N20 is in grid four; also I have used a translation of (0/2). Therefore I expect the formula to be
T2 = 5N – 7G - (5 × 6 × 2):
T in grid 3 = 118
T in grid 4 = 58
Therefore the formula here is:
T2 = 5N – 7G - 60
This again means that I was correct in my prediction.
Therefore overall this means that the general formula for the horizontal translation of the t-shape (0/y) is: T2 = 5N – 7G – (5 × G × y) which can also be expressed as
T2 = 5N – 7G – 5Gy. Also here N is the t-number of the original t-shape.
Boundaries for N:
I have noticed that there is a connection between G and the places where N can go, this is that for the highest and most right in the grid N can go the connection is
3G – 1. And for the highest and most left N can go the connection is 2G + 2. To prove that this works I will use G6:
(3 × 6) – 1 = 17
This shows it to be correct, because if the t-shape was any higher or further right then there would not be 5 numbers in the t-shape.
(2 × 6) + 2 = 14
This shows it to be correct, because if the t-shape was any higher or further left then there would not be 5 numbers in the t-shape.
The difference in these shows the maximum translation:
17 – 14 = 3
Therefore the maximum translation is (G- 3/0). Now I have found the general formulas for a horizontal and vertical translation, I will try to find the overall formula for any translation. The way to combine the vertical and horizontal translations is by adding them:
(x/0) + (0/y) = (x/y)
T 2= T + 5x -5Gy
T2 = 5N – 7G – 5Gy + 5x
However this is only my prediction therefore I shall test this twice, to make sure that I can make a valid conclusion:
Firstly I will use G6; and use the values of N being 34 and 20. N34 will be in grid 5 and N20 will be in grid 6. Therefore the translation is (-2/2).
T = (5 × 34) – (7 × 6) + (5 × -2) – (5 × 6 × 2)
T = 58
Now I will add up the five numbers in the translated shape and if they equal 58, then I can say that the formula works in this case:
20 + 14 + 7 + 8 + 9 = 58
Therefore this shows that the formula is correct.
Now I shall test my general formula on G7, with the translation of (-3/2), with the t-number being 41. the grids that I shall use is grid 3 and 4
T = (5 × 41) – (7 × 7) – (5 × 7 × 2) + (5 × -3)
T = 71
Now again I will add up the five numbers in the translated t-shape:
24 + 17 +10 + 9 +11 = 71
Again the they prove to be the same.
Now I can accurately say that the general formula for any translation is:
T2 = 5N – 7G – 5Gy + 5x
Proof for the formula:
When plotting the original shape we notice that there is an algebraic connection of-
Then if the t-shape is translated (x/y), there is an algebraic pattern inside the translated t-shape:
Now I shall add these five terms up:
N – 2G -1 + x – Gy
N – 2G + x – Gy
N – 2G + 1 + x – Gy
N – G + x – Gy 5N – 7G +5x – 5Gy
N + x - Gy
This is the same as the general formula that I found out, which means that I can make a very valid conclusion, saying that this is the general formula.
Restrictions and boundaries:
As in previous accounts, there are restrictions of the values x and y can take:
When the t-shape is at the highest and most in the corner that it can get (as in G10, grid 3), making sure that there are still five numbers I the t-shape. It can only be translated horizontally at (7/0) , as in grid 4, and the furthest that it can be translated vertically is (0/7) ,as in grid 5 as well. When one looks at this there is a connection between G and values x and y can take. This is that for maximum translation for x is
3 – G ≤ x ≤ G – 3
The maximum translation for y is the same 3 – G ≤ y ≤ G – 3 this is because the grids are squares, however if this was not the case, then this would not be true.
I will now prove this by testing them on random grid sizes:
G7, I will use grid 5, with the original t-shape to have N16.
From what I have deduced I can say that horizontally the maximum translation can only be (4/0) or (-4/0). And the vertical translation can only be (0/4) or (0/-4).
And I am correct, this is because the translations are (4/0) and (-4/0).
Effect of rotation 90º clockwise on the t-shape.
Here I will try to find the formula that connects the original t-shape to a rotation of about (c/d). I will use T3 as the new shorthand for the t-total
Firstly I have rotated a t-shape about point (0/0), in G7 in grid 6.
T for the rotated shape is 162.
However I have also found out that there is a connection between T and N, which is:
This shows that the formula for a rotation about (0/0) is T3 = 5N + 7. I worked this out by adding all the algebraic terms in the rotated t-shape.
Now I will do a rotation of (1/0). I will do this on G10 in grid 4
T for the rotated shape is 177
However again I have also found out that there is a connection between T and N, which is:
This shows that the formula for a rotation of (1/0) is T3 = 5N – 5G + 12. I worked this out by adding all the algebraic terms in the rotated t-shape.
Now I will do a rotation of (2/0). I will do this on G10 in grid 5
T for the rotated shape is 182
However again I have also found out that there is a connection between T and N, which is:
This shows that the formula for a rotation of (2/0) is T3 = 5N -10G + 17. I worked this out by adding all the algebraic terms in the rotated t-shape.
Now I will do a rotation of (3/0). I will do this on G10 in grid 6
T for the rotated shape is 187
However again I have also found out that there is a connection between T and N, which is:
This shows that the formula for a translation of (3/0) is T3 = 5N – 15G + 22. I worked this out by adding all the algebraic terms in the rotated t-shape.
I can now see a pattern emerging for (c/0). It is:
T3 = 5N + 7+ 5c – 5cG
I will now test this, by firstly using the formula to find T for the rotated shape and then I will manually add up the T for the rotated shape.
I will use G6 in grid 7 to rotate the t-shape (-1/0):
Using the formula:
T = (5 × 14) + (7) +(5 × -1) – (5 × -1 ×6)
T = 102
Now adding the numbers in the rotated shape:
T = 19 + 20 + 21 + 27 +15
T = 102
This shows that the formula of T3 = 5N + 7 + 5c – 5cG is correct.
Proof for my formula:
Now to get the formula from this I shall add up the algebraic terms in the translated shape:
T = (N+ c – cG) + (N+ c – cG + 1) + (N + c – cG + 2) + (N + c – cG + 2 – G) +
(N + c – cG + 2 + G)
T3 = 5N + 7 + 5c – 5cG
I can say that a rotation 90º clockwise about (c/0) is identical to a rotation 90º clockwise about (0/0), followed by a translation (c/c).
As I know this, it also confirms the result of T3 = 5N + 7 + 5c – 5cG:
Rotation 90º clockwise about (0/0) => 5N + 7
Followed by a translation (c/c) => 5c – 5cG
Also this idea shows identical as the general formula of part two
Rotation 90º clockwise about (0/0) => 5N + 7
Followed by a translation (x/y) => 5x – 5yG
Now I will do rotation of 90º clockwise about a point (0/y) from the N
As before when the centre of rotation is (0/0) the formula is T = 5N + 7.
Firstly I have rotated the t-shape about point (0/0), in G9 in grid 7.
T for the rotated shape is 199.
However I have also found out that there is a connection between T and N, which is:
This shows that the formula for a vertical rotation of (0/0) is T3 = 5N + 7. I worked this out by adding all the algebraic terms in the rotated t-shape.
Next I have rotated the t-shape about point (0/1), in G6 in grid 8.
T for the rotated shape is 72.
However I have also found out that there is a connection between T and N, which is:
This shows that the formula for a vertical rotation of (0/1) is T3 = 5N – 5G + 2. I worked this out by adding all the algebraic terms in the rotated t-shape.
Now I will rotate the t-shape about point (0/2), in G9 in grid 8.
T for the rotated shape is 182.
However I have also found out that there is a connection between T and N, which is:
This shows that the formula for a vertical rotation of (0/2) is T3 = 5N – 10G - 3. I worked this out by adding all the algebraic terms in the rotated t-shape.
Now I will rotate the t-shape about point (0/3), in G6 in grid 10.
T for the rotated shape is 133.
However I have also found out that there is a connection between T and N, which is:
This shows that the formula for a vertical rotation of (0/3) is T3 = 5N – 15G - 8. I worked this out by adding all the algebraic terms in the rotated t-shape.
Form these I have found that there is a pattern. Therefore I predict that the general formula for (0/d) is T3 = 5N + 7 – 5dG – 5d. I will now test this with a 90º rotation clockwise about (0/-2) it will be tested on G9 on grid 9.
The t-total for the rotated shape, using my predicted general formula is:
T = (5 × 43) + (7) – (5 × -2 × 9) – (5 × -2)
T =322
Now I will manually add up the five numbers inside the t-shape and if the sum of this is identical to T using the predicted formula then I can say that it works.
T =322
Proof for the formula
Now I will add up all the algebraic term inside:
N – dG - d
N – dG - d + 1
N – dG - d + 2 – G
N – dG - d – 2 5N – 5dG – 5d + 7
N – dG - d + 2 + G
This is the same as the general formula that I found out, which means that I can make a very valid conclusion, saying that this is the general formula.
Again I can say that a rotation 90º clockwise about (0/d) is identical to a rotation 90º clockwise about (0/0), followed by a translation (-d/d).
As I know this, it also confirms the result of T3 = 5N + 7 + 5c – 5cG:
Rotation 90º clockwise about (0/0) => 5N + 7
Followed by a translation (-d/d) => + - 5d – 5dG
Therefore: T³ = 5N + 7 – 5d – 5dG
Therefore overall I can say that for a 90º clockwise rotation about point (c/d), will be the same as a 90º clockwise rotation about point N, (0/0), followed by a translation (c/c) and then followed by a translation (-d/d). I will now test this on G6 on grid 7 and 8.
I have used method 1, in grid 7, of doing a rotation of 90º about point (3/-2). The new rotated t-shape’s N is 47. Then I have done a rotation of 90º about point (0/0), in grid 8, (this is in pink) then I have done a translation of (3/3) (in green) this is to cater for (c/c). Then I have done a translation of (2/-2) (in red) this is to cater for (-d/d). from this I have found that the 90º clockwise rotation about point (c/d), will be the same as a 90º clockwise rotation about point N, (0/0), followed by a translation (c/c) and then followed by a translation (-d/d) has N of 47.
Therefore the rotated shapes both have N of 47, this corresponds. Therefore I can say that I was correct.
From this I can find a general formula from this. To do this I will take the translation of (c/c) which is 5N + 5c – 5cG and then add this to the translation of (-d/d) which is 5N + 7 + 5d – 5cG. This is the equivalent of 5N + 7 + 5c – 5cG – 5d – 5dG, this formula is the general formula for a 90º clockwise rotation followed by
T3 = 5N + 7 + 5c – 5cG – 5d – 5dG.
I will now test this on G10 on grid 8. Firstly I will do a rotation of 90º about point (3/2) and add up the numbers inside the rotated shape:
T = 37 + 38 + 39 +29 + 49
T = 192
Now I will apply the formula that I worked out, and if the outcome of this equals 192 then the general formula for any translation followed by a 90º clockwise rotation is correct.
T = (5 × 42) + (7) + (5 × 3) – (5 × 3 × 10) – (5 × 2) – (5 × 2 × 10)
T = 192
This shows that the formula that I predicted is correct.
Finally I will investigate the effect of a translation (x/y), followed by a rotation of 90º clockwise (c/d) from the new t-number.
I can say that the original t-shape has the formula of T = 5N – 7G, this is because as I found I part one, the formula of a t-shape in any grid size is T = 5N – 7G, then if this shape is translated (x/y), then it will have the formula of T = 5N – 7G + 5x – 5Gy, I know this because I found this in the first section. Then I shall rotate the t-shape 90º clockwise about the point (c/d), away from T of the translated shape. This is also the same as a rotation 90º clockwise about point (0/0), followed by a translated of (c/c)and after this followed by another translation of (-d/d).
I will now test this on G6 on grid 11. I will use the original N of 20, then I will translate the shape (0/1) then the translated shape, I will rotate 90º about point (1/0). From this I will find T and see if there is a relationship between c, d, x, y, G and N:
T = 42
This is the same as: 1) translation (0/1) 2) now rotate 90º clockwise
3) translation (1/1)
If the t-shape is translated (x/y)
From this if there is a rotation 90º clockwise about N + - Gy is:
By adding these terms in the diagram I will be finding the new total:
T = (N + x – Gy) + (N +x – Gy + 1) + (N + x – Gy + 2) + (N + x – Gy + 2 – G)
+ (N + x – Gy + 2 + G)
T = 5N + 5x – 5Gy + 7
Now if there is a translation of (c/c) I can say that:
T = 5N + 5x – 5Gy + 7 + 5c – 5cG
Then if there is a translation of (-d/d) I can say that:
T = 5N + 5x – 5Gy + 7 – 5d – 5dG
If these are then combined for a translation (c/d) then:
T = 5N +5x – 5Gy + 7 +5c – 5d – 5Gc – 5Gd.
However this is only my prediction, therefore I will now test this by using the formula first to find out the t-total of the rotated and translated shape, and then I will manually add up the five terms inside the rotated and translated shape and if the results both comply then the formula must be correct. I will test this on G10 on grid 10, the original t-shape will have N54 then this will be translated (-2/-1) then this shape will be rotated from the point (2/-1):
T = (5×54) + (5×-2) – (5×10×-1) + 7 + (5×2) – (5×-1) – (5×10×2) – (5×10×-1)
T = 282
Now I will add the five terms inside the rotated shape from the translated shape, and if the sum of this equals 282, then the formula works:
T = 55 + 56 + 57 + 47 + 67
T = 282
This means that the overall formula for a translation (x/y), followed by a rotation of 90º clockwise (c/d) from the new t-total is T = 5N + 5x – 5Gy + 7 + 5c – 5d – 5Gc – 5Gd.
Evaluation:
Therefore overall from my investigation, I have found that:
- For a translation (0/y) the general formula is T = 5N – 7G – 5Gy
- For a translation (x/0) the general formula is T = 5N – 7G + 5x
- For a translation (x/y) the general formula is T = 5N – 7G – 5Gy + 5x
- For a rotation 90º clockwise about point (c/0) T = 5N + 7 + 5c – 5cG
- For a rotation 90º clockwise about point (0/d) T = 5N + 7 – 5dG – 5d
- For a rotation 90º clockwise about point (c/d) T = 5N + 7 – 5dG – 5d – 5Gy + 5x
- For a translation (x/y),followed by a rotation of 90º clockwise from point (c/d) from the translated t-shape, T = 5N + 5x – 5Gy + 7 + 5c – 5d – 5Gc – 5Gd
However, due to time restriction I could only find the effects of a 90º clockwise rotation, but if time was not of the essence, then I could find the effect of a 180º rotation and 270º rotation and see if the is a connection between them all.
