T no = 20
T = 1 + 2 + 3 + 11 + 20 = 37
T no = T
T = T-19 + T-18 + T-17 + T-9 + T = 5t – 63
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
T-total = T-19 + T-18 + T-17 + T-9 + T
= 5T-63
= 5t-number-63 = t-total.
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. I 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
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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 being nine. If the grid size were 10 by 10 then it would be 10*7. At the end of this piece of coursework when I put all the formulas together I will realise that the number we minus or plus by is divisible by 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.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
If I 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
I will now be 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 will be finding out more about the grid size and what it is capable of doing.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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.
Now 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.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
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 2nd rule were you times the grid size by seven.
11/05/05 T – Totals Prashant Sawlani
Coursework
Workings:
The 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.
T-number is 13
T-total = 19+29+39+21+13 = 121
5tn+56= t-total
5*13+ 56= 121
11/05/05 T – Totals Prashant Sawlani
Coursework
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 D
5tn+63 = t-total U
5tn-7= t-total R
The different size of grid changes means the formula has to change slightly.
This is what happened.
