T=32
X=5x32-63 (13+14+15+23+32=97)
X=160-63
X=97
T=71
X=5x71-63 (52+53+54+62+71=292)
X=355-63
X=292
I have tested the formula for three different T-numbers and the formula works for a 9 by 9 grid.
The full formula for this size grid is: - X=5T-63
2) The T-Shape on different size grids, translated to different positions to Investigate relationships between the T-Total, the T-Numbers and the Grid sizes.
Below is a grid with parts of the formula in. I will use this in a minute to figure out a formula for any T-Total on any size grid.
Grid Width (= G), and the T-Number (= T), the T-Total (= X)
For a “ T ” in the grid
T-total (X) = (T-2G-1)+(T-2G)+(T-2G+1)+(T-G)+(T)
X=5T-7G
If the Grid Width (= G), and the T-Number (= T), the T-Total (= X) is calculated by
X=5T-7G
I will now try the formula on an 8 by 8 grid.
The formula is X=5T-7G
Which changes to X=5T-7x8
X=5T-56
T. Number T. Total
18 34
23 59
63 259
58 234
1+2+3+10+18=34
6+7+8+15+23=59
41+42+43+50+58=234
46+47+48+55+63=259
X=5x18-56 (1+2+3+10+18=34)
X=90-56
X=34
I will now try it for two more of my t-shapes to make sure the formula works correctly.
X=5x23-56 (6+7+8+15+23=59)
X=115-56
X=59
X=5x63-56 (46+47+48+55+63=259)
X=315-56
X=259
I will now try the formula on a 7 by 7 grid.
The formula is X=5T-7G
Which changes to X=5T-7x7
X=5T-49
T. Number T. Total
16 31
20 51
44 171
48 191
1+2+3+9+16=31
5+6+7+13+20=51
29+30+31+37+44=171
33+34+35+41+48=191
X=5x16-49 (1+2+3+9+16=31)
X=80-49
X=31
I will now try it for two more of my t-shapes to make sure the formula works correctly.
X=5x20-49 (5+6+7+13+20=51)
X=100-49
X=51
X=5x48-49 (33+34+35+41+48=191)
X=240-49
X=191
I will now try the formula on a 6 by 6 grid.
The formula is X=5T-7G
Which changes to X=5T-7x6
X=5T-42
T. Number T. Total
14 28
23 73
33 123
1+2+3+8+14=28
10+11+12+17+23=73
20+21+22+27+33=123
X=5x14-42 (1+2+3+8+14=28)
X=70-42
X=28
I will now try it for two more of my t-shapes to make sure the formula works correctly.
X=5x23-42 (10+11+12+17+23=73)
X=115-42
X=73
X=5x33-42 (20+21+22+27+33=123)
X=165-42
X=123
I have tried this formula out on three different grid sizes and the formula works, the formula is: X=5T-7G, when Grid Width (= G), the T-Number (= T), and the T-Total (= X)
3) Grids of different size using other transformations and combinations of transformations to Investigate relationships between the T-Total, the T-Numbers, the Grid sizes and the transformations.
Below is a grid with parts of the formula in. I will use this in a minute to figure out a formula.
For a “ T ” in the grid
2 squares right
1 square down
From 2G+2 to 3G+4
This looks like adding numbers of squares down to the number of G’s and the number of squares right to the other number. If we start with a T-number (T) then moving 4 right and 3 down
The new T-number is:
T+3G+4
We can use “h” for the number of places right and “Y” for the number of places down.
We could write this as (h/y), a column vector:
Now X+(4/3)=X+3G+4
And X+(h/y)=X+yG+h
For all of the “ T’s ” the T-number is the same as shown below:
The other “ T’s ” that I can work out the formula for are shown bellow.
A 9 by 9 grid can be used for the other 3.
T. Number T. Total
- 142
41 198
41 212
41 268
22+23+24+32+41=142
30+39+48+40+41=198
34+43+52+42+41=212
58+59+60+50+41=268
I will now try to find out the formula for my t-shape when the T is 90° from it upright position when (T=T. Total) and (G=grid size).
All of the formulas in the grid above added together gives me an overall formula of:
T+(T+1)+(T+2-G)+(T+2)+(T+2+G)
5T +7
Now that I know the formula for the T-Shape 90° from it upright position, I will now test it 3 times, with different sized grids.
I will test my new formula, in an 8 by 8 grid.
5T + 7
=5x27 + 7 (T. Total: 21+29+37+28+27=142)
=135 + 7
=142
I will now test my formula again, on a 7 by 7 grid.
5T + 7
= 5x17 + 7 (T. Total: 12+19+26+18+17= 92)
= 85 + 7
=92
I will now test my formula on a 6 by 6 grid.
5T + 7
= 5x14 + 7 (T. Total: 10+16+22+15+14= 77)
= 70 + 7
=77
I will now try to find out the formula for my t-shape when the T is upside down (180° from it upright position) when (T=T. Total) and (G=grid size).
All of the formulas in the grid above added together gives me an overall formula of:
T+(T+G)+(2G+T-1)+(2G+T)+(2G+T+1)
7G+5T
Now that I know the formula for the T-Shape upside down, I will now test it 3 times, with different sized grids.
I will firstly try it on an 8 by 8 grid.
7G+5T
=7x8+5x29 (T. Total: 44+45+46+37+29=201)
=56 + 145
=201
A 7 by 7 grid.
7G+5T
=7x7 + 5x18 (T. Total: 31+32+33+25+18=139)
= 49 + 90
= 139
Also, now on a 6 by 6 grid.
7G+5T
=7x6 + 5x15 (T. Total: 26+27+28+21+15=117)
= 42 + 75
= 117
I will now try to find out the formula for my t-shape when the T is 260° from it upright position when (T=T. Total) and (G=grid size).
All of the formulas in the grid above added together gives me an overall formula of:
T+(T-1)+(T-2+G)+(T-2)+(T-2-G)
5T-7
Now that I know the formula for the T-Shape 260° from it upright position, I will now test it 3 times, with different sized grids.
I will test my new formula, firstly on a 9 by 9 grid.
5T-7
=5x30 – 7 (T. Total: 19+28+37+29+30=143)
=150 – 7
=143
I will now try it on an 8 by 8 grid.
5T – 7
=5x29 – 7 (T. Total: 19+27+35+28+29=138)
=145 – 7
=138
I will now try it on a 7 by 7 grid.
5T – 7
=5x40 – 7 (T. Total: 31+38+45+39+40= 193)
=200 – 7
=193
Whilst doing this investigation I found out that for the 2 T’s which were sideways like this:
Had the formula’s 5T – 7 and 5T + 7 which did not have a G (Grid Size) in the formula, because they would cancel each other out thus telling me that this formula was not affected by the grid size in order for me to find the T. Total with my formula’s.
