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T-Total and T-Number

PART 1

We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below: -

The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted will be called the t-total. In this section there is an investigation between the t-total and the t-number.

For this t-shape the

T-number is 20

And the

T-total is37

For this t-shape the

T-number is 21

and the

T-total is 42

As you can see from this information is that every time the t-number goes up one the t-total goes up five.

Therefore the ratio between the t-number and the t-total is 1:5

This helps us because when we want to translate a t-shape to another position. Say we move it to here

We all ready know the answer to the one in red. To work out the one in green all we have to do is work out the difference in the t-number and in this case it is 54. We then times the 54 by 5 because it rises 5 ever time the t- number goes up. Then we + the t-total from the original t-shape and we come out with the t-total for the green t-shape. This is another way to work out the t-total.

            What we need now is a formula for the relationship between the t-total and the t-number. I have found a formula which is 5t-number-63 = t-total.

            The question is how did we work out this formula and what can we do with it?

The formula starts with 5* the t-number this is because there is a rise in the t-total by 5 for every t-number. We then –63 which do by working out the difference between the t-number and another number in the t-shape. This has to be done to the other 4 numbers in the t-shape. Here is an example: -

The t- shape has a t-number of 32. Now to work out the difference between the t-number and the rest of the numbers in this t-shape

Working out: -

32-13=19

32-14=18

32-15=17

32-23= 9

TOTAL= 63

This will happen to all the shapes this way up. To prove this I will do another.

The t-number is 70. Now to work out the difference between the t-number and the rest of the numbers in this t-shape

Working Out: -

70-51=19

70-52=18

70-53=17

70-61=9

TOTAL=63

Again the number turns out to be 63. This is where the 63 came from in this equation. There is also another place this 63 comes from. This is 9*7=63. The nine in this comes from the size of the grid this one been nine. If the grid size were 10 by 10 then it would be 10*7. At the end of this piece of coursework when we but all the formulas together we realise that the number we minus or plus by is divisible b y seven. This is where we get the seven from. The seven works with all the same sizes. The other method will also work with a different size grid.

If we add these two together we have our formula.

                                                5tn-63=t-total

Here is an example of using the formula

5*57-63=t-total

5*57-63= 222

Check

T-total = 38+39+40+48+57=222

This formula has proven to work.

'font-size:14.0pt; '>PART 2

This next section involves using grids of different sizes and then translating the t-shape to different positions. Then investigation of the relationship between the t-total, the t-number and the grid size. Here we are doing what we did in the last section but finding out more about the grid size and what it is capable of doing.

T-total = 1+2+3+13+24 = 43

T-number = 24

The t-total and the t-number have risen even though the t-shape looks to be in the same place. The t-number has risen by four and the t-total has risen by six. If we use the same rules we made in the last section it works. Here is the longer method

Difference

24-1= 23

24-2 = 22

24-3 =21

24-13 =11

TOTAL =77

Or the shorter way

7* 11 (grid size) = 77

Try out the new formula

5tn – 77= t-total

5*24-77=43

The same formula works with only changing the last number in the formula. This will be tried on a smaller grid size to make sure it is not if the grid size is bigger.

T-number = 10

T-total = 1+2+3+6+10= 22

7 * 4 (grid size) = 28

5tn- 28= t-total

5*10-28=22

This has proven to work on a smaller scale. We can see that by changing the grid size we have had to change the formula but still managing to keep to the rule of how you get the number to minus in the formula.

PART 3

In this next section there is change in the size of grid. Also there is transformations and combinations of transformations. The investigation of the relationship between the t-total, the t-numbers, the grid size and the transformations.

            If we turned the t- shape around 180 degrees it would look like this. When we have done this we should realise if we reverse the t-shape we should have to reverse something in the formula.

 It is obvious that we will have to change the minus sign to a different sign. We should try the opposite of minus which is plus

5tn + 63=t-total

5 * 2 + 63 = 73

Check to see if the formula has worked

T-number = 2

T-total = 2+11+19+20+21 =73

The reverse in the minus sign has worked. 

The next step is to move the shape on its side. Again we nearly keep the same formula as we had at the beginning. Again we change the minus number. We can work out the number to minus by working out the difference in the t-number to each number in the t-shape.

Difference

12-1 =11

12-10= 2

12-19= -7

12-11 = 1

TOTAL = 7

Formula

5tn - 7 =t-total

5*12 - 7= 53

Check to see if the formula is right

T-number = 12

T-total = 1 +10 +19 +11 +12 = 53

This formula has worked. If we rotated the t-shape 180 degrees, The same will happen, as what happened when the t-shape was turned 180 degrees from it is first original position. This is proven below.

5tn + 7 = t-total

5* 70 + 7 = 357

Check

T-number = 70

T-total = 70+71+72+63+81 = 357

If we were to put the t-shape diagonally on the grid we find that the same rule applies again apart from you can not use the 2^nd rule were you times the grid size by seven.

The red t-shape has t-number of 33 and the t-total = 7+17+27+25+33 = 109

The difference between the t-number and the rest of the numbers in the t-shape.

33-25= 8

33-7= 26

33-17= 16

33- 27 = 6

TOTAL= 56

5tn+56= t-total

5 * 33 - 56 =109

The reverse triangle the sign should be reversed to a plus. The t-shape used here is the one in blue.

T-number is 13

T-total = 19+29+39+21+13 = 121

5tn+56= t-total

5*13+ 56= 121

The t-shapes above holds more formulas as the rest they all work the same.

The red t-shape has a t-number of 32 and a t-total of 32+42+52+60+44= 230

This t-shape has a formula the formula is 5tn + 70 = t-total

To see if this formula works

First we work out the difference in between the t-number and the rest of the numbers in the t-shape.

Difference

42-32= 10

52-32= 20

60-32= 28

44-32= 12

TOTAL= 70

5*32 + 70 = 230

The blue shape is the opposite of the red t-shape so therefore the formula for the blue t-shape is 5tn – 70 = t-total. The sign has become the opposite of what it use to be. This has happened in many cases before.

Now that we have worked out all the formulas for the position in the normal sized t-shape. We can try enlarging the t-shape. If we double the t-shape (volume is four times bigger). The grid below shows the new shape. I have added all the numbers together in the squares of the t-shape. This leaves us with our original t-shape but with larger numbers in the grid.

The t-number turns out to be 176. This is the bottom four numbers added together. The t-total is 356. I have worked out the differences between the t-number and the rest of the t-shape.

Difference

176-24 = 153

176-32 = 144

176-40 = 136

176-84 = 92

TOTAL= 524

Now we have the rest of the formula. The formula is very much the same apart from the number we minus or plus by is vaster.

Formula

5tn – 524 = t-total

5*176-524 = 356

Formula has proven to work.

We have seen that there is a relationship with all the transformations made to the t-shape. Everything that we have done the t-shape has seemed to link to the part that was discovered before. These still stays the same apart from we add an extra part on to the end of the formula. This is because we are not looking for a link between all the positions of the t-shape when it is a certain way up. Here we want to find out whether there is a link between only two t-shapes. Here first of all we are looking for a link when we rotate this t-shape 90 degrees.

Here we have t-shapes with the same t-number. Now we want a formula for rotating a t-shape 90 degrees. We already have two separate formulas. The red t-shapes formula is 5tn- 63= t-total. The blue t-shapes formula is 5tn + 7= t-total. If we add the 63 and the 7 together from the two formulas we get 70. This is the difference in the t-total between the two t-shapes. The t-number for both t-shapes is 41. The red t-shape t-total is 142. The blue t-shape t-total is 212.

If we keep our original formula which is 5tn - (7 * grid size)

Then we add the difference in the t-shapes t-total and we get this

5tn - (7*9) + 70 = t-total

5*41-63+ 70 = 212

The formula has worked. We now want to work out the difference in the t-total of the first t-shape we started with to the rest of the other six t-shapes. The next two are the below t-shapes.

The blue t-shapes t-total is a difference of 126 to the original t-shape that had a t-total of 142.

Formula

5tn – (7*G) + 126 = t-total.

5*41-(7*9) + 126 = 268

The red t-shape therefore will be

5tn – (7*G) + 56 = t-total

5*41- (7*9)+ 56 = 198

The next four t-shapes are just the same apart from you – the (7*G)

Red t-shape

5tn- (7*G)+7= t-total

5*41 – 63+7 = 149#

Blue t-shape

5tn- (7*G) + 119 = t-total

5* 41 –63+ 119 =261

The last two t-shapes

Red t-shape

5tn- (7*G) + 133 = t-total

5* 41 –63+ 133= 275

Blue t-shape

5tn- (7*G) -7 = t-total

5*41-63-7 = 135

W now have a formula for seven different rotations. The number at the end of the formula we plus by or in one case minus buy again are divisible by seven. You could say that the magic number for this piece of coursework is seven. Like they have a magic number in the bible that is 12.

            If there are formulas for rotation then surly there is for reflection. Here I have simply only done one type of reflection just to prove that reflection actually works. Here is the formula 5tn+ (12gm) = t-total. How do we get this formula is what we need to know.

The answer to this is that you need to think of what you are doing to each of the numbers in the t-shape from the blue t-shapes t-number. For the number 29 we have a grid movement of one so we get (tn+gm). For the number 38 we have a grid movement of two so we get (tn+2gm). For the numbers 46, 47 and 48 we have a grid movement of three and a total of three numbers, se we get 3(tn+3gm). The total of all of them together is (5tn +12*gridsize) = t-total.

This formula should be tested. The t-total of the blue t-shape is 37 and the t-total of the red t-shape is 208.

Formula

5tn+(12*gridsize)= t-total

5*20+ 12* 9 = 208

The formula has worked.

CONCLUSION

 In this project we have found out many ways in which to solve the problem we have with the t-shape being in various different positions with different sizes of grids. The way we have made the calculations less difficult is by creating a main formula that changes for all the different circumstances.

Here I have put all the formulas I have come up with. These formulas only apply to the nine by nine grids

5tn-63= t-total     D

5tn+63 = t-total     U

5tn-7= t-total      R

5tn+7= t-total     L

5tn-70= t-total     DR

5tn+70 = t-total     UL

5tn-56= t-total     DL

5tn+56 = t-total     UR

The different size of grid changes means the formula has to change slightly.

This is what happened.

 We also have formula for rotation, which are

We have a formula for reflection which is 5tn+(12*gridsize)= t-total.

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