Working out
32-13=19
32-14=18
32-15=17
32-23=9
19+18+17+9=63
This is where 63 comes into the equation 5n-63
I will also use another method to check the formula is correct
n+n-23+n-19+n-18+n-17= 5n-63
This is an example of using this formula
5x48-63=t-total
5x48-63=177
Check to prove formula
T-total=29+30+31+39+48=177
This proves my formula works.
I will now try using grids of different sizes and translating the t-shape into different positions. I will then investigate the relationship between the t-number, t-total and grid size.
T-number=22
T-total=40
The t-number and t-total have both risen even though the t-shape is in the same place as it was before. The t-number has risen by 2 and the t-total by 3. This is because I am now using a ten by ten grid. I will now try and find some more t-numbers and t-totals for some more t-shapes on this grid.
T-number=55
T-total=34+35+36+45+55=205
Now I will try and find a formula for the ten by ten grid. I will first find the difference between the t-total and other numbers in the t-shape.
22-1=21
22-2=20
22-3=19
22-12=10
Total=70
Here is another way to figure out the formula
n+n-10+n-20+n-21+n-19
So this would mean that my formula will be 5n-70
I will try and test my new formula
5n-70=t-total
5x22-70=40
The same formula can be used for each grid with only changing the last number in the formula. I will now try it on a different sized grid to prove this.
T-number=18
T-total=1+2+3+10+18=34
Difference
18-1=17
18-2=16
18-3=15
18-10=8
Total=56
n+n-8+n-17+n-16+n-15= 5n-56
5n-56=t-total
5x18-56=34
As you can see by changing the grid size we have to change the end of the formula but we still keep the rule of how you get the number to minus in the formula.
So I can find formulas for all grid sizes, which are shown below in the table
Looking at my results I can clearly see that the numbers at the end of the formula are 7x the grid number.
From my investigation I can now say that to find the t-total for any size grid. The rule is 5n-7g (g stands for grid width)
I will now try and rotate the t-number 180 degrees. I will then try and find a formula for this rotation. I know that having done this the t-shape has been reversed so this would mean something in the formula would also have reversed.
T-number=2
T-total=2+11+19+20+21=73 (the reverse of 37 is 73)
I will now try and find a formula.
The difference between each t-total is 5 so my equation will again start with 5, I will then multiply 5 with 2 because it is the first t-number. Also because I am using a 9x9 grid I know that the equation will end with 63.
So the formula will be 5n+63=t-total
To check and make sure the formula works we do:
5x2=10+63 = 73
The minus sign has been reversed to a plus sign.
n+n+9+n+17+n+18+n+19=5n+63
I will now rotate the t-shape again 90 degrees to 270 degrees and try and find a formula for 270 degree rotation of the t-shape.
T-number=12
T-total= 1+10+19+11+12=53
I will now try and find a formula
The difference between each t-total is 5 so again the formula will start with 5, I will then multiply 5 with 12 the t-number. 5x12=60 we then minus 7 to get 53 the t-total.
So therefore the formula will be:
5n-7
To check and make sure the formula works we do:
n+n-1+n-2+n-11+n+7=5n-7
I will now rotate the t-shape 180 degrees from its last position. I predict that the same will happen as before and the equation will stay the same but the minus sign will change to a plus sign. I will prove this below.
T-number=10
T-total= 10+11+12+3+21=57
To find the formula 5n+7 that I predicted we do:
5x10(t-number) =50+7=57
To check the formula we do:
n+n+1+n+2+n+11+n-7=5n+7
The formula is correct.
Conclusion
In this coursework I found many ways to solve the problem with the t-shape being in various different positions and different grid sizes. I have made my calculations less difficult by creating a formula which changes according to the different circumstances.
Below I have shown the different formulas I have come up with these only apply to the 9x9 grid.
5n-63
After 180 degrees rotation
5n+63
270 degrees rotation
5n-7
90 degrees rotation
5n+7
