OR
5xN-63
I have the T-Total but I want to prove my formula by testing out two more T-Numbers. I am doing this because I want to make sure that my rule is accurate and I want to prove that my rule does work in any T-Numbers.
N= 32 N=26 T=13+14+15+23+32=97 T=7+8+9+17+26=67
5x32-63=97 5x26-63=67
I have found out that my rule is correct. I know this because the rule matches with the T-Total. I have also found out that if I put the T-Shape anywhere in the grid it works and that proves that my rule was correct.
Rule in Algebra:-
My aim was to find the T-Number for each of the numbers above. The first thing that I did was to find out the T-Number, which is eventually at the bottom of the T-shape. Then I subtracted the T-Number by the other numbers inside the T-Shape, for example, I subtracted 65 from 74, which gave the answers N-9+N-18+N-17+N-9+N, and this is how I found the T-Number for the entire number grid. I have noticed that the middle column is going up in nine and I think this is because of the size of the grid. I think that in every other grids which I would do the T-Number is going to gain on the grid size and go up in multiples e.g. 9 by 9 grid, the T-Number is going to go up in nine’s. I concentrated on the number 74 because it was the T-Number. This is the formula for the T-Total= N-19+N-18+N-17+N-9+N= 5N-63
Now I am going to test my theory to see if it works with other T-Numbers.
Key: - N= T-Number
T= T-Total
T =1+2+3+11+20=37 T=16+17+18+26+35=112 T-Total= N-19+N-18+N-17+N-9+N=5N-63 T-Total=N-19+N-18+N-17+N-9+N=5N-63 T=5x20-63=37 T= 5x35-63=112
I have found out the formula 5N-63 works anywhere in the grid, I know this because I tested the formula two times for different T-Numbers and the formula works. The full formula for the nine by nine grid is: - T=5N-63
Now I am going to find out by using different grids the rule in algebra, I am doing this because I want to find out does the grid size matter and if it does how. I am going to do an eight by eight grid to start.
The formula for the T-Total is:-
T-Total = N-17+N-16+N-15+N-8+N=5N-56
T= 5x35-56=119
Now I am going to test this formula to see if it works
Key: - N= T-Number
T= T-Total
T =9+10+11+18+26=74 T=30+31+32+39+47=179 T-Total= N-17+N-16+N-15+N-8+N=5N-56 T-Total= N-17+N-16+N-15+N-8+N=5N-56 T= 5x26-56=74 T= 5x47-56=179
I have found out that the formula 5N-56 works anywhere in 8 by 8 grid, I have tested the formula two times and this proves that my result does not have in faults in it. The formula for the 8 by 8 grid is: - T=5N-56
Now I am going to do a seven by seven grid to find out the Algebraic rule
The formula for the T-Total is:-
T-Total = N-15+N-14+N-13+N-7+N=5N-49
T= 5x16-49=31
Now I am going to test this formula to see if it works
Key: - N= T-Number
T= T-Total
T=11+12+13+19+26=81 T=17+18+19+25+32=111
T-Total = N-15+N-14+N-13+N-7+N=5N-49 T-Total = N-15+N-14+N-13+N-7+N=5N-49
T= 5x26-49=81 5x32-49=111
I have found out the formula 5N-49 works anywhere in the 7 by 7 grid, I know this because I tested the formula two times for different T-numbers and the formula works. The formula for the 7 by 7 grid is: - T=5N-49
I am now going to do a six by six grid to find out the Algebraic rule
The formula for the T-Total is:-
T-Total = N-13+N-12+N-11+N-6+N=5N-42
T= 5x34-42=128
Now I am going to test this formula to see if it works
Key: - N= T-Number
T= T-Total
T=8+9+10+15+21=63 T=13+14+15+20+26=88
Total = N-13+N-12+N-11+N-6+N=5N-42 T-Total = N-13+N-12+N-11+N-6+N=5N-42
T= 5x21-42=63 T= 5x26-42=88
I have found out that the formula 5N-42 works anywhere in 6 by 6 grid, I have tested the formula two times and this proves that my result does not have in faults in it. The formula for the 6 by 6 grid is: - T=5N-42
This is my last and final grid.
The formula for the T-Total is:- T-Total = N-21+N-20+N-19+N-10+N=5N-70 T= 5x95-70=405
Key: - N= T-Number
T= T-Total Now I am going to test this formula to see if it works
T=57+58+59+68+78=320 T=12+13+14+23+33=95 T-Total =N-21+N-20+N-19+N-10+N=5N-70 T-Total =N-21+N-20+N-19+N-10+N=5N-70 T= 5x78-70=320 T= 5x33-70=95
I have found out that the formula 5N-70 works anywhere in 10 by 10 grid, I have tested my theory two times and I found out that my rule works and this proves that my result does not have any faults in it. The formula for the 10 by 10 grid is: - T=5N-70
This is all the Algebraic rules, which I found when I was investigating the T-Total and the T-Number.
Overall, the general rule for the T-Total is 5N-7G, where the N is T-Number and the G is grid number.
I recognised that if I multiply the number of grids by 7 I would get the T-Total.
General Rule: 5N-7G
Now I am going to test my general rule with two examples and then prove that I was right:-
General rule in algebraic form
now I will add the numbers to find the General rule, N-2G-1+N-2G+N-2G+1+N-G+N I got the N-G because the middle square is the same as the grid number.
Using my rule that I found out I predict that if the grid size is 5 by 5 the rule is: 5N-35 Now I am going to prove it 5N- 7G is 5N-7x5, so the rule is 5N-35.
Now I am going to predict and prove it for a 4 by 4 grid just to make sure that my general rule is correct. Therefore, the rule for a 4 by 4 grid is going to be 5N-28. Now I am going to prove it; 5N-7G is 5N-7x4, so the rule is 5N-28.
I have predicted and proved it twice to see if my general rule is 5N-7G and by my prediction, I have proved that my rule is correct.
