I will call the T-number Tn
Tn=60
Tn + 41 + 42 + 43 +51 = T-total
=237
I have shown by this line of algebra that the 5 numbers added together equals the T-total (the T-number plus the other 4 numbers in the T-shape). Which equals 237.
I will now substitute in the formulae the difference between the T-number and the other numbers in the T-shape:
Tn + Tn-19 + Tn–18 + Tn-17 + Tn-9 = 237
I have changed the formulae into a simplified one by finding the difference between each number in the T-shape and putting it into the equation.
I have simplified it to 5Tn – 63 = T-total
This is the final formula.
I will now test out to see if my formula is correct by choosing a random T-number and seeing if it gives the correct T-total
5Tn-63 = T-total
5x80-63 = 337
I have substituted the T-number into the equation
My formulae is correct as if I were to do the sum manually without using the formulae by adding 61+62+63+71+80 (all the numbers in the T-shape) it will equal 337 the same answer I got using my formulae. This proves that my formulae Is correct.
Part 2 – Use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total, the T-numbers and the grid size.
To investigate the relationship between the T-total, T-number and the grid size I will need to put the template into algebraic form.
I have changed the template into algebraic form. When I added the equations in the T-shape (Tn + Tn-g + Tn-2g-1 + Tn-2g + Tn-2g+1) and got the answer 5Tn – 7g, where Tn is the T-number and g is the grid size.
I will now try and see if the formulae are correct for different grid sizes.
I will put the T-number into the formulae with a 6x6 grid
5Tn-7g
5x27 – 7x6 = 93
If I were to do the sum manually by just adding the numbers in the T-total it would be:
14+15+16+21+27 = 93
This is the same number as I got with the equation. I will try this once more to confirm that my formulae are correct.
I will put my T-number into the equation
5Tn-7g
5x83 – 7x11 = 338
If I were to do this manually: 60+61+62+72+83 = 338
The same result as if I were to do it with the formulae this proves that the formulae are correct.
To investigate translation of the T-shape to different positions
Part 3 – Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-total, T-numbers, the grid size and the transformations.
I will try and find a formulae for rotation of the T-shape I will draw my 9x9 grid again to try and find out this formulae.
In my grid I have chosen a random T-number of 40 and will find a formulae for rotation
Tn-19 + Tn-18 + Tn-17 + Tn-
= 9 + Tn
= 5Tn-63
Tn + Tn+1 + Tn+2 + Tn-7 +
= Tn+11 = 5Tn+7
Tn + Tn+9 + Tn+17 +
= Tn+18 + Tn+19
= 5Tn+63
= Tn-11 + Tn-2 +Tn+7 + Tn-1
+ Tn
= 5Tn-7
I can see from this that:
When the T is at 0° the formulae will be 5T-7g
When the T is at 90° the formulae will be 5Tn+7
When the T is at 180° the formulae will be 5Tn+7g
When the T is at 240° the formulae will be 5Tn-7
I will now test my formulae to see if they are correct by using the formulae I found and seeing if I get the same results as I would doing it manually.
0° 90°
5Tn-7g 5Tn+7
5x40 – 7x9 =137 5x40+7 =207
180° 240°
5Tn+7g 5Tn-7
5x40 + 7x9 =263 5x40-7 = 193
Now I will do the calculations manually:
0° 90°
Tn=40 Tn=40
Tn+21+22+23+31 = 137 Tn+41+42+33+51 =207
180° 240°
Tn=40 Tn=40
Tn+49+57+58+59 =263 Tn+39+38+47+29 =193
I got the same results as I did by using my formulae this proves that my formulae is correct for rotation.