This pattern occurs because each number inside the T goes up one as you move it across the grid and there are five numbers in the T. This pattern also works for every T on the grid.
The next thing I did was name each of the squares in relation to the T-Number
The N stands for the T-Number and the other numbers are in relation to it and it works for every T. This helps because it makes it easier for me to work out a piece of algebra to solve the problem for every T.
In this T I have noticed that the first difference from N is 9 which is also the width of the grid.
I’ll put that idea into another T. Note W= Width Number (9)
This is the same thing as before but shown algebraically.
If you add all of these together the answer would be
5N-(7W+1-1) = T-Total
In this formula there is a +1 and a -1 which even each other out is if you do this the formula would be
5N-7W = T-Total
Aim 2-Different Sizes and Relationships
I know this formula works for the 9 by 9 grid but I’m not sure if it will work for any other grids.
I have highlighted the first T that can be made on this grid. The T-Number is 22. If the formula works for this T then the T-Total should be 40. The T-Total is 1+2+3+12+21= 40.
This formula works on that T but does it work on any other T’s in the grid.
This T’s T-Number is 69. If the formula works for this T the T-Total should be 275. The T-Total is 48+49+50+59+69= 275.
The formula seems to work for this size grid.
As there are 10 in each row it’s obvious that the row above will be 10 less than the row below. So 59 is 10 less than the T-Number 69. If you calculate the whole T you realise that row 2 is 10 less than row 1 and row 3 is 20 less than row 1, but there are three relevant numbers in row 3 which are 19 less, 20 less and 21 less that the T-Number. The 19 and the 21 can both be changed into 20’s because if you take the 1 from 21 and put it on the 19 they both make 20.
Row 3
Row 2
Row 1
So finally on row 2 there is -10 and on row 3 there is -60. If you add these together they make -70 or -7W. W stands for Width.
The principle is the same for this grid except instead of the rows going up in 10’s they go up in 12’s so
The -25 and -23 can both be changed into -24’s.
So on row 2 there is -12 and on row 3 there is -72. If you add them up they make -84 or -7W.
Aim 3- Transformation and its effects on the formula
I will do this with a 12 by 12 grid to start with, as this will give me enough accuracy to start with.
This T will be called ST 64 as it starts at 64, it is a stretch of 2 in both directions.
ST 64=26+27+28+29+30+40+52+64= 296
I think I can work out the formula using my previous method so:
12+24+ (36x5) =216
Now if I see how many times 12 goes into 216
216/12 = 18
This means the formula is:
8N-18W= T-Total
8N= 8=number of integers in the T shape N= T-Number
18W= 18= Number calculated W=Width
Just to make sure this formula works here is another T.
ST 142 = 104+105+106+107+108+118+130+142= 920
Using the formula the T-Total would be (8x142) + (18x12) = 920
You would be able to use this way of finding out the formula for any T.
Conclusion
In conclusion I have found that the number of integers in the T calculates the number before the N in the formula and the width of the grid calculates the value of W. The number before the W is found by looking at the numbers in the rows and relating them to the T-Number. When the T is regular the W-Number is negative e.g.
8N
Whereas if the T is flipped upside down the W-Number is positive.