46+47+48+56+65= 262
T-Number=65
T-Total=262
61+62+63+71+80=337
T-Number=80
T-Total=337
For 9x9 Grid:
- When the T-Number is even, the T-Total is odd.
- When the T-Number is odd, the T-Total is even.
I will now find a rule which links the T-number with the T-Total:
n+(n-9)+(n-18)+
(n-18)+(n-19)
=5n-63
When n=44 =(5x44)-63=157
Testing
25+26+27+35+44=157
8x8 Grid
1+2+3+10+18= 34
T-Number=18
T-Total=34
20+21+22+29+37= 129
T-Number=37
T-Total=129
33+34+35+42+50= 194
T-Number= 50
T-Total= 194
45+46+47+54+62= 254
T-Number= 62
T-Total= 254
For 8x8 Grid:
- When the T-Number is odd, the T-Total is odd.
- When the T-Number is even, the T-Total is even.
I will now find a rule which links the T-number with the T-Total:
n+(n-8)+(n-16)+
(n-18)+(n-17)
=5n-56
When n=36 =(5x36)-56=124
Testing:
19+20+21+28+36=124
As you can see my rule has worked.
T-Totals – Any sized Grid
I will now find the general rule for any sized grid, which links the T-Number with the T-Total.
n+(n-G)+(n-2G)
+(n-2G-1)+(n-2G+1)
= 5n-7G
When n=65, and G=10 =(5x65)-(7x10)= 255
44+45+46+55+65= 225
As you can see my rule has worked.
Translation:
If I translate the T 3 Vectors right, it will become:
22+23+24+33+43= 145 25+26+27+36+46= 160
T-Number=43 T-Number=46
T-Total= 145 T-Total=160
- The T-Total has increased by 15.
- This because there 5 numbers in the T-Total, which all have increased by 3
- 5x3=15
-
Consequently, if the vector is () the formula for moving T across would be: T-Total+5A
If I translate T, 3 vectors up, it will become:
62+63+64+73+83= 345 32+33+34+43+53= 195
T-Number=83 T-Number=53
T-Total= 345 T-Total=195
- The T-Total has decreased by 150.
- This is because, there are 5 numbers in the T-Total, which all have been decreased by 3 lots of grid number. (3G) In this example the grid size is 10.
-
Consequently if the vector is () the formula for moving the T up would be: T-Total-5bG
- Therefore, if the T moves across by ‘a’ each square increases by ‘a.’
- Therefore, if the T moves up by ‘b’ each square increases by ‘b.’
The formula before moving the T is:
n+(n-G)+(n-2G)+(n-2G-1)+
(n-2G+1)= 5n-7G
Consequently, the formula for moving the T, by any vector () is:
(n+a-bG)+(n-G+a-bG)+(n-2G+a-bG)
+(n-2G-1+a-bG)+(n-2G+1+a-bG)
=5n-7G+5a-5bG
- This shows that if T is translated across by ‘a’ each square increases by ‘a’.
- This shows that if T is translated up by ‘b’ each square decreases by ‘bG.’
There may be limitations to translation, for example translating might take the T-Shape, out of the grid.
Is this a problem?
6x6 Grid
The T-Total for this T is
Then translating the by vector ( ), this goes outside the grid.
5n-7G+5a-5bg
(5x28)-(7x6)+(5x3)-(5x2x6)= 53
But if we imagine the T-shape continues onto the next lines and we use the formula we have, we get the same answer as the T-Total of
= 53
So, there is not a problem, because the rule still works. Actually there aren’t any limitations, we just simply imagine the T-shape continues onto the next lines.
Rotation from n
I will now find the general rule for rotating T on point n.
90*
n+(n+1)+(n+2)+(n+2-G)+
(n+2+G)= 5n+7
180*
n+(n+G)+(n+2G)+(n+2G-1)+
(n+2G+1)= 5n+7G
270*
n+(n-1)+(n-2)+(n-2+G)+
(n-2-G)= 5n-7
Rotation on any point
I will now find the general rule for rotating T from any fixed point, which is not ‘n’. I will describe that point as a vector from ‘n’. Vector written in the form ( ) .
90*
= n+3-2G-2-3G = formula for new T-number
Replacing numbers with letters:
= n+c-dG-d-cG = formula for new T-number
5n+7 is the formula for T rotated at 90* on point n.
I will now substitute the old ‘n’ (T-number) with the new ‘n’ (the new T-number).
5(n+c-dG-d-cG) +7 5n+5c-5dG-5d-5cG+7
This is the new formula for the T-Total of my newly rotated shape.
180*
= n+3-2G-2G+3 = Formula for new T-number
Replacing numbers with letters:
= n+2c-2dG = Formula for new T-number
5n+7G is the formula for T rotated at 180* on point n.
I will now substitute the old ‘n’ (T-number) with the new ‘n’ (the new T-number).
5(n+2c-2d) +7G 5n+10c-10dG+7G
This is the new formula for the T-Total of my newly rotated shape.
270*
= n+3-2G+2+3G = Formula for new T-number
Replacing numbers with letters.
= n+c-dG+d+cG = Formula for new T-number
5n-7 is the formula for T rotated at 270* on point n.
I will now substitute the old ‘n’ (T-number) with the new ‘n’ (the new T-number).
5(n+c-dG+d+cG) -7 5n+5c-5dG+5d+5cG-7
This is the new formula for the T-Total of my newly rotated shape.
Combination (Rotation & Translation)
- I will now find the general rule for rotating and then translating a T.
- The rule will only work in certain Grids, because the vector by which I want to translate the T, will be too big to fit the limitations of the Grid.
- The limitations are only a problem, if you want the T-shape to stay in the grid, but I have demonstrated already using the 6x6 grid that you can imagine the numbers carrying on.
- To do this I must combine the rules for rotation with the ones for translations:
90*= 5n+5c-5dG-5d-5cG+7+5a-5bG
This allows you to rotate the T by 90* and then move it by any vector.
180*= 5n+10c-10dG+7G+5a-5bG
This allows you to rotate the T by 180* and then move it by any vector.
270*= 5n+5c-5dG+5d+5cG-7+5a-5bG
This allows you to rotate the T by 270* and then move it by any vector.
