T-number69-50+48+49+59+69=275(t-total)
T-number32-11+13+12+22+32=90(t-total)
As you can see from this table every time the t-number goes up 1 the t-total goes up 5. From this I can tell that to find formula connecting these two together I will have to multiply one of them by 5. So then I decided to multiply the t-number by 5.
This table shows that 5 times the t-number minus 70 equals the t-total.
So the formulae relating the t-number and the t-total is-
T-number=N T-total=T
T=5N-702) Different sizes and relationship between the T-total and the T- number and the grid size
I know this works for the grid 10 by 10 but I'm not exactly sure if it'll work for any other grids because there are more intervals in between the first number of each line. So I might have to change a value in my formula. I will try my formula with two different sized grids.
Here is a 9 by 9 grid for which I will test my formula
T-number20-1+2+3+11+20=37(T-total)
I notice this is 3 less than my first result that I got for my 10 by10 grid. I predict this is because there are different values such as in my 10 by 10 grid the vertical numbers were 2, 12, and 22 but in my 9 by 9 grid the vertical numbers were 2, 11, 20. If we add these numbers together they come to 10by 10 grid-36 and I the 9 by 9 grid- 33. Here I can see where the difference comes from.
Now I will use my formula to see if it works for this grid. I will use a random t-number.
T=5N-70
T= (5x50)-70
T=250-70
T=180
Now I will check my answer
T=31+32+33+41+50
T=187
Here I can see I have two different answers. I think should try using 7x grid width instead of 70.
Grid Width=W
T=5N-(7W)
T=250-(7x9)
T=250-63
T=187
My formula was correct in a sense but not a 100% accurate. My formula now is-
T=5N-7W
Here is a 5 by 5 grid for which I will test my ‘new’ formula
There aren’t many results to be had from a 5 by 5 table. There are 9 to be exact.
I will pick another random number -
T-number18-7+9+8+13+18=55(T-total)
So I have picked the t-number18 I will try to find the t-total using my ‘new’ formula and see if I get the same answer as above.
T=5N-7W
T= (5x18)-(7x5)
T= 90-35
T=55
From my test I have shown my method is correct.
My method seems to have worked out as it is logical and fairly straight forward to explain.
My formula now after being changed does shows the relationship between the T-number, T-total, and the grid size (in my case just the width)
3) Relationships between the T-total and the T-number and the grid size and the transformations
I am going to use a simple yet relatively small grid to investigate the relationship between the T-total and the T-number and the grid size and the transformations. I am now going to investigate the effect on the T-Total when the T-Shape is transformed, rotated 90°, both T-Shapes will have the same T-Number like this:
T- Number = 12
Orange T-Total = 12+1+2+3+7=25
Gold T-Total = 12+13+14+9+19=67
I will see if my formula can be used in these two cases.
T=5N-7W
T=60-35
T=25
The orange t-total is correct according to my formula.
T=5N-7W
T=60-35
T=25
The gold t-total is not correct according to my formula. It has given the same answer as the orange t-total.
This shows that I need to try something else. I will next investigate what I need to do to my formula to make it work for transformations. First I will try finding the difference between the 2 different t-totals.
Gold-Orange=42
This shows that I have to add 42 to the formula for my relationship to work.
So now my formula will become- T=5N-7W+42
Next I am going to use a simple yet relatively small grid to investigate the relationship between the T-total and the T-number and the grid size and the transformations. This time I am going to use a 6 by 6 grid. I am going to do the same thing for this grid as I did for the last.
T- Number =20
Orange T-Total = 7+8+9+14+20=58
Gold T-Total = 20+21+22+28+16=107
Orange-Gold=49
This shows that I have to add 49 to the formula for my relationship to work.
So now my formula will become- T=5N-7W+49
Next I am going to use a 7 by 7 grid
T- Number =23
Orange T-Total = 8+9+10+16+23=66
Gold T-Total = 23+24+25+32+18=122
Orange-Gold=56
This shows that I have to add 56 to the formula for my relationship to work.
So now my formula will become- T=5N-7W+56
I am going to show my results in a table.
Conclusion
My formula’s had to be changed each time I started the next task. Which I found confusing at first but then I ‘got into it’.
For my final formula I have made it to be
Where T=T-total, N=T=number, W=Grid width
T=5N-7W
For example-
The t-number=26 in 7 by 7 grid
T=(5x26)-(7x7)
T=130-49
T=81
Then if there’s a transformation-
T= (5N-7W)+((W+1)x7)
For example-
The t-number=25 in 8 by 8 grid and then rotated 90 degree’s clockwise.
T= (5N-7W)+((W+1)x7)
T=(125-56)+(9x7)
T=169+63
T=232
