I will now use my formula to work out the T-Total
T-Total = 5N – 63
= 210 - 63
= 147
This shows that my formula works as the numbers in my T added up = 23+24+25+33+42 = 147.
So the overall rule for an upright T in a 9X10 grid would be 5N-63.
I am now going to work out the rule for a T in the upright position for a 10X10 grid. I am going to work it out by doing the same thing as a 9X10 grid. Working out 5 Ts that are next to each other, making a table and then predicting the 6th. Then I will work out the formula for a 10X10 grid.
T-Number - 22
T-Total – 40
T-Number - 23
T-Total – 45
T-Number – 24
T-Total – 50
T-Number – 25
T-Total - 55
T-Number – 26
T-Total – 60
From this table I can see that each time I move the T one place to the right, this is because each digit increases by 1, making it 5 each time. From this I predict that for T-Number 27 the T-Total will be 65, I will now prove my theory.
T-Number - 27
T-Total – 65
This shows that my prediction was correct, each time I move the T one place to the right that the T-Total increases by 5. This is because each time the T moves one place to the right each number increase by 1, there are 5 numbers in the T, so the T-Total will increase by 5.
The formula for a T in the upright position in a 10X10 grid.
I am going to use the T with the numbers
From this I can see that there is a rule for the way that the numbers in the T are arranged and I can see that it is
From this I can see that on a 10 by 10 grid the formula equals
N + (N-10) + (N-20) + (N-21) + (N-19)
I can now simplify this by gathering up all the n terms and the numbers to make the formula 5N – 70
I will now prove this on a 10 by 10 grid using the numbers
I will now use my formula to work out the T-Total
T-Total = 5N – 70
= (57X5) – 70
= 285 - 70
= 215
This shows that my formula works as the numbers in my T added up = 36+37+38+47+57 = 215
So the overall rule for an upright T in a 10X10 grid would be 5N-70
Working out the algebra to an upright T
The algebra to a T is simply the difference between the numbers in the 2.
In every grid it will always be the same. It will be the T-Number at the bottom, then the T-Number - the grid number to give you the second number.
The 3rd number would be the T-Number – the grid number times two.
The number on the left would be T-Number – the grid number times 2 + 1.
The number on the right would be T-Number – the grid number – 1.
If I put thins into a T it would look like this
Working out a constant rule in all grid sizes
For the 9X10 and 10X10 grid sizes I have found out an easy way to find out the overall rule for how to work out a T in the upright position. It is to find out the algebra of the difference in the number in the T. This is what I mean by the difference in the numbers in the T in the T below
Difference of 10
Once you work out the difference between all of the numbers in the Y you can start to see a pattern.
In a 10X10 grid the difference between the T-Number and the number above it will always be the T-Number – the grid number. So for a 10X10 grid it would be T-Number – 10.
For the 2nd number above the T-Number it would be the T-Number – 2 lots of the grid number. On the 10X10 grid it would be T-Number – 20.
For the number, one place to the right it would be the T-Number X 2 +1. So on a 10X10 grid it would be T-Number – 21.
Then on the number to the left it would be the T-Number – 2 lots of the grid number, takeaway 1. On this T it would be T-Number – 19.
Once you have found out the algebra you can then work out a formula for the grid size. You do this by drawing out a T like I have done above and put in the difference between the numbers, then gather up all the numbers and all the “N”s for the gird above it would
T-Total = N + (N-10) + (N-20) + (N-19) + (N-21)
Then gather up all the Ns and the total would be 5N-70.
This is the way to work out the formula for a grid number. I am now going to work out the formula for an upright T in the grid sizes 7X10, 8X10, 11X10 and 12X10.
7X10
Using these numbers I can now work out the difference between each number and then work out the formula for a 7X10 grid. The difference between each number is below.
Now I have worked out the difference between the numbers I can now gather up the Ns and the numbers to give me the formula
5N-49.
So the rule for an upright T in a 7X10 grid would be 5 X T-number – 49. I will now prove my theory works.
So with the T-Number being 18 the formula would be
5 X 18 – 49 = 41. To show my theory works I will now add the number up by hand
3 + 4 + 5 + 11 + 18 = 41
8X10
Using these numbers I can now work out the difference between each number and then work out the formula for an 8X10 grid. The difference between each number is below
Now I have worked out the difference between the numbers I can now gather up the Ns and the numbers to give me the formula
5N – 56
So the rule for an upright T in an 8X10 grid would be 5 X T-number – 56. I will now prove my theory works.
So with the T-Number being 77 the formula would be
5 X 77 – 56 = 329. To show my theory works I will now add the number up by hand
60 + 61 + 62 + 69 + 77 = 329
11X10
Using these numbers I can now work out the difference between each number and then work out the formula for an 11X10 grid. The difference between each number is below.
Now I have worked out the difference between the numbers I can now gather up the Ns and the numbers to give me the formula
5N – 77
So the rule for an upright T in an 11X10 grid would be 5 X T-number – 77. I will now prove my theory works.
So with the T-Number being 41 the formula would be
5 X 41 – 77 = 128. To show my theory works I will now add the number up by hand
18 + 19 + 20 + 30 + 41 = 128
12X10
Using these numbers I can now work out the difference between each number and then work out the formula for a 12X10 grid. The difference between each number is below.
Now I have worked out the difference between the numbers I can now gather up the Ns and the numbers to give me the formula
5N – 84.
So the rule for an upright T in a 12X10 grid would be 5 X T-number – 84. I will now prove my theory works.
So with the T-Number being 41 the formula would be
5 X 102 - 84 = 426. To show my theory works I will now add the number up by hand
77 + 78 + 79 + 90 + 102 = 426
Conclusion
I have now found out the formulas for a T in an upright position in a 7,8,9,10,11,12 X 10 grid. Using the difference of the numbers from the T-Number in the gird. I will now put all these formulas into one table and see if I can notice a pattern, then predict what a 13X10 formula would be.
I am now going to prove my theory on a 13 X 10 grid
5 X 28 – 91 = 49
1 + 2 + 3 + 15 + 28 = 49
This proves that my theory works, each time you go up a grid size you increase the number you subtract from 5N by 7, I can also see that the number you subtract from 5N is 7 lots of the grid size. So from this I can conclude that the overall formula for an upright T in any grid would by 5N – 7G.
I will prove my formula by using a 24X10 grid
Using my formula 5N – 7G I can work out that
(5 X 50) – (7 X 24) = 82
1 + 2 + 3 + 26 + 50 = 82
This shows that my formula 5N – 7G works.
Rules for T Rotated
I am now going to work out the rule for a T in a 9X10 and 10X10 grid on different angles, the angles I am investigating are below
I am going to use the method I worked out before, using the difference between the numbers in each T.
I am going to start by working out a T upside down in a 9X10 grid. The T number will always stay at the bottom of the T, no matter which way up it is. The T-Number is the box in red
Using the method I used before, by working out the differences I am now going to work out the formula for the T.
T-Total = N + (N+9) + (N+18) + (N+17) + (N+19)
From this I can see that the total is
5N + 63,
To prove this work I will now try it out.
(5 X 23) + 63 = 178
23 + 32 + 41 + 40 + 42 = 178
This shows that my formula 5N + 63 works for an upside down T in a 9X10 grid.
I am now going to work out a T on its side in a 9X10 grid. To do this I am going to use the same method as above, I am going to work out the difference between the numbers in the T, then I will gather up all the numbers and the Ns to work out my formula.
Using these numbers I am going to work out the difference between the each number in the T, I will then put it into a diagram below.
This diagram shows me the difference between the numbers in any 9X10 grid that is on its side, now I have the differences I am going to gather them up to get my formula.
T-Total = N + (N+1) + (N+2) + (N-7) + (N+11)
Now I will gather it all up and it comes to 5N + 7.
To prove this works I am going to randomly select a T from a 9X10 grid then work out its T-Total by adding it up, and then using my formula.
T-Total = 5N + 7
= (5X59) + 7
= 295 + 7
= 302
By using the formula my T-Total is 302 I will now prove this by adding the numbers up manually
59 + 60 + 61 + 52 + 70 = 302.
This shows that my formula for a T on its side is 5N + 7. I am now going to work out the formula for a T on its other side.
Using these numbers I am going to work out the difference between the each number in the T, I will then put it into a diagram below.
This diagram shows me the difference between the numbers in any 9X10 grid that is on its side, now I have the differences I am going to gather them up to get my formula.
T-Total = N + (N - 1) + (N - 2) + (N + 7) + (N - 11)
Now I will gather it all up and it comes to 5N - 7.
To prove this works I am going to randomly select a T from a 9X10 grid then work out its T-Total by adding it up, and then using my formula.
T-Total = 5N – 7
= (5X62) – 7
= 310 – 7
= 303
By using the formula my T-Total is 303 I will now prove this by adding the numbers up manually
62 + 61 + 60 + 51 + 69 = 303.
This shows that my formula for a T on its side is 5N - 7.
I have now found out the formulas for Ts in a upright, upside down and on bots side positions I am now going to put them into a table below and try to work out a pattern.
