The T-number for this T-shape is 21.
Using the same method I have used above, I believe if I add all the 5 numbers in the T-shape I would come up with a T-total of 42.
2+3+4+12+21=42
42 is the T-total inside this T-shape. Correct
Using my examples I have made up a table to show that T-total goes up in 5 whenever the T-number shifts along a column.
Prediction
Prediction
Prediction
As you can see in the table above every time you minus the T-number and the T-total together the answer goes in order of 4.
Proof that my result works:
The T-number for this T-shape is 22.
Using the same method, I believe if I add all the 5 numbers in the T-shape I would come up with a T-total of 47.
3+4+5+13+22=47
47 is the T-total inside this T-shape. Correct
The T-number for this T-shape is 23.
Using the same method, I believe if I add all the 5 numbers in T-shape I would come up with a T-total of 52.
4+5+6+14+23=52
52 is the T-total inside this T-shape. Correct
The T-number for this T-shape is 24.
Using the same method, I believe if I add all the 5 numbers in the T-shape I would come up with a T-total of 57.
5+6+7+15+24=57
57 is the T-total inside this T-shape. Correct
Formula
In this table I have times the T-number by 5. I have used the number 5 because there are 5 numbers in the T-shape. The answer, using that method was the taken away by the T-total that I had found out by adding up all the numbers in the T-shape. So every time I used the equation I have proved that the formula 5N-63=T-total.
To make sure that my finding of the 9x9 grid formula is correct I am going to use a random T-shape.
T-number=65
5x65=325
325-63=262
46+47+48+56+65=262
Therefore the T-total=262 Correct
I have proven that my formula 5N-63=T-total does work.
Expressing result algebraically:
T-total= N+ (N-9) + (N-18) + (N-17) + (N-19) =5N-63
Part 2
I am now going to experiment on using different size grids as well as investigating relationships between the T-total, the T-numbers and the grid size.
Firstly I am going to use an 8x8 grid, reason being that I am going to start off with small grids then work up to larger grids.
8x8 Grid
Below I have chosen my examples of T-shapes that I am going to experiment with.
The T-total of the numbers in this T-shape is:
1+2+3+10+18=34.
The T-number is 18.
The T-total of the numbers in this T-shape is:
2+3+4+11+19=39.
The T-number is 19.
The T-total of the numbers in this T-shape is:
3+4+5+12+20=44.
The T-number is 20.
The T-total of the numbers in this T-shape is:
4+5+6++13+21=49.
The T-number is 21.
The T-total of the numbers in this T-shape is:
5+6+7+14+22=54.
The T-number is 22.
The T-total of the numbers in this T-shape is:
6+&+8+15+23=59.
The T-number is 23.
Using my examples I have worked out that a pattern is starting to form.
Prediction
Every time a T-number goes up by 1, The T-total goes up by 5.
The T-total of the numbers in this T-shape is:
35+36+37+44+52=204.
The T-number is 52. Correct
The equation for an 8x8 grid is 5N-56.
N+ (N-8) + (N-16) + (N-15) + (N-17)
5N-56=T-total. Correct
Proof that the equation I have worked out by using a number of examples works.
5x52-56=204.
35+36+37+44+52=204.
The T-total is 204. Correct.
Now I am going to use a larger grid size, 10x10
10x10 Grid
Below I have chosen my examples of T-shapes that I am going to experiment with.
The T-total of the numbers in this T-shape is:
1+2+3+12+22=40.
The T-number is 22.
The T-total of the numbers in this T-shape is:
4+5+6+15+25=55.
The T-number is 25.
The T-total of the numbers in this T-shape is:
7+8+9+18+28=70.
The T-number is 28.
In my 3 T-shapes I have discovered that a pattern is starting to form.
Down below I have done a table to show the sequence in which the T-number and T-total is starting to form.
Each time the T-number goes up in +3, the T-total starts to go up in +15.
The T-total of the numbers in this T-shape is:
64+65+66+75+85=355.
The T-number is 85.
The T-total of the numbers in this T-shape is:
65+66+67+76+86=360.
The T-number is 86.
Another pattern is starting to form but this time each time the T-number goes up +1 the T-total goes up +5.
Here is a table to show my finding of a second pattern forming:
In all I have found out that if the T-number went up by +1 each time the T-total would go up by +5 and if the T-number went up by +3 the T-total goes up by +15.
I have proved this above.
Expressing result algebraically:
T-total= N+ (N-10) + (N-20) + (N-21) + (N-19)=5N-70
Proof that my equation does work.
T-total=40
40=5x-70
(5x22)=110
40=110-70
110-70=40
Master equation for Part 2:
5x-(grid size x7) =T-total
Proof of Master equation for grid size 8
5x63-(8x7) =259
46+47+48+55+63=259 Correct.
Proof of Master equation for grid size 8
5x92-(10x7) =390
71+72+73+82+92=390 Correct.
Part 3
Now instead of just shifting the directions of the T-shape I am going to rotate them, by this I mean turning the T-shape upside down and moving it in opposite directions. But I am still going to have to investigate the relationships between the T-total, the T-numbers, the grid size and the Transformations.
I am going o start off with grid size 11.
11x11 Grid
Below I have chosen my examples of T-shapes that I am going to experiment with.
First of all I am going to turn my T-shapes facing left.
The T-total is:
70+78+79+80+91=398
The T-number is 78.
I was unable to finish off Part 3 due to running out of time. But if I were able to complete Part 3 I would have got examples of many T-shapes facing different directions up, down, left and right. I would have also put in many results tables and included lots of information of all sort of things, from examples to expressing results algebraically.
