T totals. In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find

 Finding relationship between T-Number and T-Total, with different sized grid and different translation, enlargements and rotations of the T-Shape

For this section we will have to keep some items constant, as for to provide a stable environment to prove of disprove theories based on translations, enlargements and rotations, I have decided to keep the number above the T-Number (x - g) constant and call it v, therefore v = (x – g)        

 

EXPLAIN WHAT THE LETTERS STAND FOR.

Preliminary work

To begin with, we shall use a 9x9 grid and keep v constant (highlighted);

The T-Total for this T-Shape is 187 (31 + 32 + 33 + 41 + 50), in relation to v these are;

If we simplify this, we can generate a formula for a relation between v and t on a grid a width of 9;

T=v + (v + 9)+(v - 9)+(v - (9 - 1) + v - (9 + 1))

T=3v+(2v-18)

T=5v-18

                        

                        We can even say that;

T=5v-2g

As 9 is the grid size, and all numbers in our relationship grid were related around 9, so we can concluded that it is related to the grid size. We can test this on different positions on the 9x9 grid.

T=(5x14) – (2 x 9)

T= 70 – 18

T=52

In addition, the T-Total by hand is;

T= 4 + 5 + 6 + 14 + 23

T= 52

Thus proving this method on a grid with of 9. We now need to try it on a grid width of 6 for example,

T = (5x16) – (2 x 6)

T = 80 – 12

T = 68

In addition, the T-Total by hand is;

T = 9 + 10 + 11 + 16 + 22

T = 68

Therefore, all the preliminary work has been completed and stated as we have found a quick method to generate the T-Total based on our new central number v. We can state that:

The T-Total (t) can be found by using the equation t = 5v – 2g, were v is the middle number

(x –g) and  g is the grid width, on any grid with, with a standard T-Shape.

We can always relate back to the T-Number at any time (the number at the bottom of the t) by making v equal to x + g, were g is the grid width and x is the T-Number, I shall use v in this question to make translations and rotations easier to achieve, as all T-Shape translations and rotations will be based on there central number, (0,0). So a practical example would be if we were to use the equation t = 5v – 2g to find the T-Total, we can use t=5(x+g)-2g, substituting v for x+g therefore relation the equation to the T-Number. This will also work in reverse, as we can find any T-Number’s value at any time by using v-g.

Translations

Vertical

Again, we shall use our standard gird size and position to establish our basic starting point;

Here we can see our basic staring point (v = 41 therefore t = 187), with a vertical translation to the second shape (v = 14 therefore t = 52). Straight away, we can generalize that,

When a T-Shape is translated vertically by a positive figure its T-Total is less than the

original T-Total

If we table these results along with all the vertical translation results from 41 to 14 (for v), we should easily see a pattern (on a grid width of 9), also adding a column for the difference between the number in that column and the once below.

MAKE A TABLE and LOOK FOR PATTERNS – TRY TO FIND A RULE

We can see an obvious relationship, that as the T-Shape is translated by +1 in a vertical direction the T-Total is larger by +45 than the previous T-Total. Therefore, we can state that,

As a T-Shape is translated vertically by +1 on a grid width (g) of 9 the T-Total (t) is

+45 larger than the previous T-Total (t) (the origin)

It is obvious we can also state that:

As a T-Shape is translated vertically by –1 on a grid (g) width of 9 the T-Total (t) is

-45 smaller than the previous T-Total (t) (the origin)

As when v = 32, t = 142 and with a translation vertically by –1 v = 23, t = 97, and 97 – 142 = -45, therefore the above statement it correct. We now need to use the same method with different grid size to make a universal equation. I have chosen a grid width of 5 as that has a central number as does a grid width of 9, but make it vertically longer as to allow room to express vertical translations. We know with will not make a difference to the final answer as proved in question 2.

As we can see we have a vertical translation of the first T-Shape (where v =23) by +3. Where v = 23, t = 105, and where v = 8, t = 30 (both found by using t = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 23 - 3).

MAKE A TABLE and LOOK FOR PATTERNS – TRY TO FIND A RULE

From this we can see that 25 is the “magic” number for vertical translations by +1 on a grid width of 5, from this I can see a link with the “magic” numbers, as for as grid width of 5 it is 25, which is 5 x 5, and for a grid width of 9 it is 45 which is 9 x 5. We can also see that translations larger than 1 can be found by a(25) (were a is the figure which you want to translate by), i.e. if you wanted to translate the T-Shape vertically by +3 , the “magic” number would be found by 5x25.

To verify this we can see what the “magic” number is on a grid width of 10, we can predict it will be 50 (5 x 10).

If we note the same form of table we have used before, we can find the “magic number”, the above graph shows a vertical translation of the T-Shape by +3, were v=46, t =210 which translates to, v=16, t=60.

As I predicted the “magic” number was 50, therefore I can generalize and state that;

Any vertical translation can be found by t=(5v-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

In terms of the T-Number (x) instead of v,

Any vertical translation can be found by t=(5(x+g)-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

Horizontal

Again, we shall use our standard gird size and position to establish our basic starting point;

Here we can see our basic staring point (v = 41 therefore t = 187), with a vertical translation to the second shape (v = 44 therefore t = 202). Straight away, we can generalize that,

When a T-Shape is translated horizontally by a positive figure its T-Total is less than the

original T-Total

If we table these results along with all the horizontal translation results from 41 to 44 (for v), we should easily see a pattern (on a grid width of 9), also adding a column for the difference between the number in that column and the once below.

We can see an obvious relationship, that as the T-Shape is translated by +1 in a vertical direction the T-Total is larger by +45 than the previous T-Total. Therefore, we can state that,

As a T-Shape is translated horizontally by +1 on a grid width (g) of 9 the T-Total (t) is

+5 larger than the previous T-Total (t) (the origin)

It is obvious we can also state that:

As a T-Shape is translated horizontally by –1 on a grid (g) width of 9 the T-Total (t) is

-5 smaller than the previous T-Total (t) (the origin)

As when v = 32, t = 142 and with a translation horizontally by –1 v = 31, t = 137, and 137 – 142 = -5, therefore the above statement it correct. We now need to use the same method with different grid size to make a universal equation. I have chosen a grid width of 11 as that has a central number as does a grid width of 9, but make it vertically shorter as we do not need the lower number as they will  not be used. We know with will not make a difference to the final answer as proved in question 2.

As we can see, we have a horizontal translation of the first T-Shape (where v =17) by +4. Where v = 17, t = 63, and where v = 21, t = 83 (both found by using t = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 21 to 17).

From this we can see that  5 is the “magic” number again as for a grid width of 9 for horizontal translations. From this an obvious relation ship can bee seen that for all grid sizes, a horizontal translation of a T-Shape by +1, makes the T-Total +5 larger, but this is only a prediction. To verify this we can see what the “magic” number is on a grid width of 10.

If we note the same form of table we have used before, we can find the “magic number”, the above table shows a vertical translation of the T-Shape by +4, were v=45, t =205 which translates to, v=49, t=225.

As I predicted the “magic” number was 5, therefore I can generalize and state that;

EXTEND THE PROJECT – EXPLORE what happens when you CHANGE THE PROBLEM in a SMALL WAY

Any horizontal translation can be found by t=(5v-2g)+5a, were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

In terms of the T-Number (x) instead of v,

Any horizontal translation can be found by t=(5(x+g)-2g)+5a, were x is the T-Number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

Combinations (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 equations, I can generate a general formula for diagonal translations, which is a prediction I need to change. One simple change needs to be made to my horizontal translation equation, that as “a” was also used for the figure by which to translate (the same as the vertical translation), we have to substitute “a” with “b” in the horizontal equation otherwise we can only move the T-Shape, is fixed diagonal positions. We need now to combine the two equations, only take one instance of 5v-2g as only one T-Shape is being translated;

WRITE YOUR RULES USING ALGEBRA

t=(5v-2g)-(a(5g))-5b

To prove this equation we need to again start with our stand grid and position and try it on a combination translation.

The above translation a combination of a horizontal equation of +3 and a vertical translation of +3 also, the origin T-Shape has a T-Total of 187, and the translated T-Shape has a T-Total of 67, using the equation we will try and generate the Second T-Total to prove our theory correct,

CHECK YOUR RULE WITH EXAMPLES

t=((5x41)-(2x9))-(3(5x9)-5x3)

t=(205-18)-(135-15)

t=187-120

t=67

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 width of 5, extended vertically to accommodate the combination translation:

Here we can see a translation of +3 vertically, and +1 horizontally, with the original T-Shape having a T-Total of 105, and the translated T-Shape having a T-Total of 35, using our generated formula we can see if it correct, if it can translate with a v total of 23 to the T-Total of translated shape of 35;

T=((5x23)-(2x5))-(3(5x5)-5x1)

T=(115-10)-(75-5)

T=105-70

T=35

CHECK YOUR RULE WITH EXAMPLES

This proves my equation correct as; the correct translated T-Total is generated, thus we can state;

Any combination translation (vertically and horizontally), can be found by

using the equation of t=(5v-2g)-(a(5g))-5b were v is the middle number,

g is the grid width, a is the number by which to translate vertically and

b is the number by which to translate horizontally.

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 canceling out that part of the equation as it would equal 0. For example if we wanted to translate a T-Shape with a v number of 41 on a grid width of 9, by –3 horizontally, the equation would look like;

T=(5x41-2x9)-0(5x9)-(5x-3)

T=(205-18)+15

T=187-15

T=172

Using our equation found in the preliminary work we can work out the real value of the translated shape, (it has a v number of 38);

T=(5x38)-(2x9)

T=190-18

T=172

Thus proving our theory right that the equation can be used for any type of translation, vertical, horizontal or a combination of the both. Thus we can state that:

Any type of translation (vertical, horizontal or a combination) can be found

by using the equation  of t=(5v-2g)-(a(5g))-5b were v is the middle number,

g is the grid width, a is the number by which to translate vertically and

b is the number by which to translate horizontally.

I terms of T-Total (x);

Any type of translation (vertical, horizontal or a combination) can be found

by using the equation  of t=(x-3g)-(a(5g))-5b were v is the middle number,

g is the grid width, a is the number by which to translate vertically and

b is the number by which to translate horizontally.

EXTEND THE PROJECT – EXPLORE what happens when you CHANGE THE PROBLEM in a SMALL WAY

Rotations

Static center of rotation

To begin with, I shall try to find generalizations and rules for static rotations of T-Shapes, were the v (middle) number is the centre of rotation.  Two other variables also need to be defined the amount to rotate by (i.e. 90, 180 or 270 degrees) and the direction I which to rotate (i.e. clockwise or anti-clockwise). To begin with, we shall start on out basic 9x9 grid, but cut the vertical height, as it will not be needed.

Firstly, we can see our T-Shape (with a T-Total (t) of 52), then a rotation of our T-Shape, rotated 90 degrees clockwise, with v (14), as it’s center of rotation, this shape a T-Total of 72. 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 generalizations.

If we plot these results into a table, again it will be easier to find generalizations:

MAKE A TABLE and LOOK FOR PATTERNS and TRY TO FIND A RULE

It is hard to make any immediate generalizations bar;

When a T-Shape is rotated by 90, 180 or 270 degrees, its T-Total is larger.

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 5 for this,

MAKE A TABLE and LOOK FOR PATTERNS and TRY TO FIND A RULE

It is not obvious to see any equations from these 2 sets of results but if we break them down we can see there are 3 different equations for the 3 types of rotation, these are best shown by the explanation below.  Were the middle number (v) is taken as the center of rotation.

Tb Shapes

Firstly Tb will be easy to find, as it is a 180-degree rotation of Ta therefore the equation will be “negative”, so the equation will be (in terms of v):

t = 5v + 2g

This will be so as all numbers are greater than the T-Number as the shape extends downwards. To prove this we can put this into practice.

Working this out using the traditional method the answer is 31 (2 + 5 + 7 + 8 + 9), using the formula:

t = (5 × 5) + (2 × 3)

t = 25 + 6

t = 31

To double check we can use a different grid size.

Again using the traditional method, we get 38 (2 + 6 + 9 + 10 + 11), using the formula:

t = (5 × 6) + (2 × 4)

t = 30 + 8

t = 38

Thus proving that:

5v + 2g can be used to find the T-Total (t) of a Tb shape, on any Grid size, regular (e.g. 5x5) or irregular (e.g. 5x101), with the two variables of grid width (g) and the Middle Number (v).

In terms of T-Number (x);

5x + 7g can be used to find the T-Total (t) of any Grid save, regular (e.g. 5x5) or irregular (e.g. 5x101), with the two variables of grid width (g) and the T-Number (x) for a Tb T-Shape.

Tc and Td shapes

As we know that a 180-degree flip will be a “negative” equation of the other flip, we only need to work out one kind of shape, therefore we shall work out formulas for Tc shapes.

Again using the “old” method we get a T-Total of 27, we have to start from scratch to make a formula so we can follow the steps used to find the original formula for Ta shapes.

t = 5 + (5 - 1) + (5 – 2) + (5 + 1) + (5 + 4)

So if we substitute 5 for v we get:

t = v + (v -1) + (v – 2) + (v + 1) + (v + 4)

t = 2v-1 + 3v + 3

t = 5v + 2

Therefore:

On a 3x3 grid the formula 5v + 2 can be used to work out the T-Total (t), were

v is the Middle Number.

The T-Total for this middle number (12) is 67, using the same method of substitution:

t = v + (v -1) + (v – 2) + (v + 1) + (v + 4)

t = 2v-1 + 3v + 3

t = 5v + 2

CHECK YOUR RULE WITH EXAMPLES

We can see the formula is the same as for a 3x3 grid.

On this 4x6 grid the total for T-Shape is 57, using the formula.

t = (5 × 11) + 2

t = 55 +  2

t = 57

Thus proving that:

The formula t = 5v + 2 can be used to find the T-Total (t) of any Tc shape

on any sized grid, were v is the Middle Number.

In terms of x;

The formula t = 5x + 7 can be used to find the T-Total (t) of any Tc shape

on any sized grid, were x is the T-Number.

Following the 180-degree flip rule (as proved by the Ta / Tb relationship) we can predict that for Td shapes the formula is t = 5v – 2.

Working this out by hand, we get a T-Total of 23 (6 + 5 + 4 + 1 + 7) for this T-Shape. If we use the formula, we get

t = (5 x 5) – 2

t = 25 – 2

t = 23

Therefore proving that:

The formula t = 5v - 2 can be used to find the T-Total (t) of any Td shape

on any sized grid, were v is the Middle Number.

In terms of x;

The formula t = 5x – 7 can be used to find the T-Total (t) of any Td shape

on any sized grid, were x is the T-Number.

Therefore, we can state from a Standard T-Shape position the following evasions can be used to generate the T-Total from v.

We can also draw up a grid for Anti-Clockwise rotations.

Combinations of translation and rotation

EXTEND THE PROJECT – EXPLORE what happens when you CHANGE THE PROBLEM in a SMALL WAY

For this section, we can predict the formula will firstly have to find the new Middle Number (of the translated shape) then that new v number will have to be put through the equation for rotations, the first part of the equation is

t=(v+b)-ag

Saying if we want the rotate the shape by 90 degrees clock wise we would use the formula t=5v+2, so our combination formula would look like;

t=5((v+b)-ag) + 2

v : is the middle number

g : is the grid width

a : is the amount to translate vertically

b : is the amount to translate horizontally

To test this we shall perform a translation and rotation of 90’s clockwise of a 9x9 grid, and see if the formula works:

On this grid, v = 41, g = 9, a = b = 3

The T-Total of the original shape is 187 and of the translated and rotated shape is 87, using the formula we get,

t=5((41+3)-(3x9)) + 2

t=5(44-27) +2

t=5(17) + 2

t=87

Thus proving my theory correct on this grid size, to double check this I will use a different grid size, and a different translation / rotation combination, in this case 180 degrees clockwise so the formula will read

t=5((v+b)-ag) + 2g

I will use an extended (vertically) grid width of 5 for this,

On this grid, v = 23, g = 5, a = 3 and b = 1

The T-Total of the original shape is 105 and of the translated shape is 55, using the formula we get,

t= 5((23+1)-(3x5)) + 2x5

t= 5(24-15) + 10

t= (5x9) + 10

t= 55

Thus proving my theory correct therefore we can state

The T-Total any combination of a translation (vertically, horizontally, or both) and a

Rotation through v, can be found by using the equation of 5((v+b)-ag) + y were v is the

Middle number is the amount to translate horizontally,  a is the amount to translate vertically, g is the grid width, and y is to be substituted by the ending required  by the type of rotation,

these are :

In terms of T-Number (x);

The T-Total any combination of a translation (vertically, horizontally, or both) and a

Rotation through v, can be found by using the equation of 5(((x-g)+b)-ag) + y were v is the

Middle number, b is the amount to translate horizontally, a is the amount to translate

vertically, g is the grid width, and y is to be substituted by the ending required  by the type of rotation, these are :

(Same as table above)

Finding a formula for non-middle number centre of rotations

In this section we will aim to find an equation to find the T-Total of a T-Shape that has been rotated, but its centre of rotation is not v, it is any integer on the grid. Here is an example below;

Here we can see a rotation of 90 Clockwise with the centre of rotation being 68, the original T-Total is 187 and the rotated T-Total is 357. We can see we will have to take the space the between the centre and the v number then rotate it, then use the 5v+2 to work the T-Total out, therefore I predict the formula will be,

WRITE YOUR RULE USING ALGEBRA

T=5(v+cg+d)+2

v = middle number

c = difference (in grid blocks) between the centre and the v number

g = grid width

d = distance between new v number and the centre of rotation

If we use this formula to generate an answer;

T=5(41+(3x9)+3)+2

T=5(71)+2

T=355+2

T=357

Thus proving this formula works, as the new T-Total is correct, but the formula is only any use if you have plotted the translation out and you know how far the new v number away is (horizontally) from the centre of rotation and it can only be used for 90 and 270-degree rotations. Thus another equation needs to be found, a more universal one.

Here we can see a rotation of 90 Clockwise with the centre of rotation being 58, the original T-Total is 187 and the rotated T-Total is 347. We need to find the difference from the original Middle number the centre of rotation then, reverse the differences (i.e. horizontal amount = vertical amount and vice versa), then use the 5v+2 method after we have found the new position of the t shape, if we do this formula is generated

T=5(c+d(g)+b)+2

EXPLAIN WHAT THE LETTERS STAND FOR

C = centre of rotation

D = horizontal difference from the relative central number

G = grid width

B = vertical difference from the relative central number

If we try this formula out;

T=5(58+1x9+2)+2

T=5(68)+2

T=347

Thus proving this formula works we now need to try it again on a different grid size with a different centre of rotation.

Here we can see a rotation of 90 Clockwise with the centre of rotation being 36, the original T-Total is 170 and the rotated T-Total is 282. Using the formula we get;

T=5(36+2x10+0)+2

T=5(56)+2

T=282

Thus proving this formula works and it is obvious that it will work in the same fashion as my static (middle number as the centre) rotations, as they both find the position as the new V number then generate the t-total based on that number, therefore I can state;

CONCLUSION – SUMMARISE WHAT YOU HAVE FOUND

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of v from the relative centre of rotation , is the grid width, and b is the vertical difference of v from the relative central number.

y is to be substituted by the ending required  by the type of rotation,

these are :

In terms of x (T-Number);

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of x from the relative centre of rotation,

g is the grid width, and b is the vertical difference of x from the relative centre of rotation.

y is to be substituted by the ending required  by the type of rotation,

these are :