The purple 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. 5n-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
5*34-63=t-total
5*34-63=107
Check
T-total = 15+16+17+25+34=107
I have just proven that my formula works
This next section involves using grids of different sizes and translating the t-shape to different positions. Then investigating the relationship between the t-total, the t-number and the grid size. Here I am doing what I 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. This is because I am now using an 11 by 11 grid. I will find the T-Total and T-number for some more T’s on this gird
T-number =52
T-Total =52+41+30+31+29=183
T-number =102
T-Total =102+91+80+79+81=433
Now I will find a formula for the 11*11 grid. First I will find the Difference between the T-Total and the other numbers in the T. then I know that it is 5n because there are still 5 squares in the T.
24-1= 23
24-2 = 22
24-3 =21
24-13 =11
TOTAL =77
So that means my formula is 5n-77
Try out the new formula
5n – 77= t-total
5*24-77=43
The same formula works for each grid with only changing the last number in the formula. This will be tried on a smaller grid size to prove that it works.
T-number = 10
T-total = 1+2+3+6+10= 22
Difference
10-1=9
10-2=8
10-3=7
10-6=4
10-10=0
9+8+7+4 =28
5n- 28= t-total
5*10-28=22
. We can see that by changing the grid size we have had to change the last part of the formula but still managing to keep to the rule of how you get the number to minus in the formula.
So I have found formulas for all grid sizes, which are
From looking at my results clearly I can see that the numbers at the end of the formulas are 7* the grid number. From this I should be able to find an overall formula that works on all grids. This formula is (g is grid width)
5n-7g
PART 3
In this next section there is change in the size of grid. Also there are transformations and combinations of T shapes. I will now be investigating 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 we have done this we should realize if we reverse the t-shape we should have to reverse something in the formula.
T-number =2
T-Total = 2+11+19+20+21=73
T-number=62
T-Total= 62+71+79+80+81=373
T-number=40
T-Total=40+49+57+58+59=263
I am now going to try and find a 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. I am using a 9*9 grid so I all ready know that the number at the end of the equation is 63.
5n + 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 but I suspect that I might have to change the sign.
T-number= 12
T-Total=1+10+11+12+19=53
T-number= 61
T-Total=50+59+60+61+68=298
T-number=50
T-Total=39+48+49+50+57= 243
Formula
5n - 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, from where it was in the last part. 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.
T-number=70
T-Total=63+70+71+72+81=357
T-number=10
T-Total-3+10+11+12+21=57
I will now find the formula
5n + 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.
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
5n+56= t-total
5 * 33 - 56 =109
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
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 (n+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(n+3gm). The total of all of them together is (5n +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
5n+(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.
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
5tn+63 = t-total
5tn-7= t-total
5tn+7= t-total
5tn-70= t-total
5tn+70 = t-total
5tn-56= t-total
5tn+56 = t-total
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.