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 I add these two together I have my 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.
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 I 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 I use the same rules I 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. I can see that by changing the grid size I have had to change the formula but still managing to keep to the rule of how I 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 I turned the t- shape around 180 degrees it would look like this. When I have done this I should realise if I reverse the t-shape I should have to reverse something in the formula.
It is obvious that I will have to change the minus sign to a different sign. I 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 I had at the beginning. Again I change the minus number. I 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 I 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 I were to put the t-shape diagonally on the grid I find that the same rule applies again apart from I can not use the 2nd rule were I 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 I 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 I have worked out all the formulas for the position in the normal sized t-shape. I can try enlarging the t-shape. If I 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 I 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.
I have seen that there is a relationship with all the transformations made to the t-shape. Everything that I have done the t-shape has seemed to link to the part that was discovered before. These still stays the same apart from I add an extra part on to the end of the formula. This is because I am 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 I am looking for a link when I rotate this t-shape 90 degrees.
Here I have t-shapes with the same t-number. Now I want a formula for rotating a t-shape 90 degrees. I 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 I add the 63 and the 7 together from the two formulas I 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 I keep our original formula which is 5tn - (7 * grid size)
Then I add the difference in the t-shapes t-total and I get this
5tn - (7*9) + 70 = t-total
5*41-63+ 70 = 212
The formula has worked. I now want to work out the difference in the t-total of the first t-shape I 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
I now have a formula for seven different rotations. The number at the end of the formula I plus by or in one case minus buy again is divisible by seven. I could say that the magic number for this piece of coursework is seven.
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 I get this formula is what we need to know.
The answer to this is that I need to think of what I’m doing to each of the numbers in the t-shape from the blue t-shapes t-number. For the number 29 I have a grid movement of one so we get (tn+gm). For the number 38 I have a grid movement of two so I get (tn+2gm). For the numbers 46, 47 and 48 I have a grid movement of three and a total of three numbers, se I 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 I have found out many ways in which to solve the problem I have with the t-shape being in various different positions with different sizes of grids. The way I have made the calculations less difficult is by creating a main formula that changes for all the different circumstances.
