I am going to find the formula by finding the diagonal difference of all the sizes within the eight by eight grid, and then try to find any patterns, which would help me in finding the formula by drawing a grid. This method would be good because it will show me the diagonal difference of any square of any size, because I would just have to insert the value of n in the formula to find out the diagonal difference of that square, therefore they will not have to write out all the squares and work out the diagonal difference.
Prediction
I predict that each time you increase the number of sides of the square within the eight by eight the diagonal difference will increase.
Method
As I progress through this investigation my aim is to find a formula or general rule which will suite or satisfy and grid size. I will do this by looking at the diagonal difference and then trying to find a pattern or a rule. I will do this by starting with the smallest grid size gradually moving on to the largest in a methodical order, and from the information, I will come up a general rule or formula. If I can, I will then convert this information into graphs and pie charts to show this information.
Results
I will first find the sequence of the diagonal difference in all of the different size grids.
Two by two grids
I will start by looking at the two by two grid.
2 times 9 = 18
10 times 1 = 10
Diagonal difference: 18-10 = 8
The diagonal difference is eight
I have found out that the diagonal difference for the two by two grid is eight, but I will try another two by two grid just to check this.
4 times 11 = 44
12 times 3 = 36
Diagonal difference: 44-36 = 8
The diagonal difference is eight again.
I again found that the diagonal difference is 8 so I know that the diagonal difference of two by two grids is 8, so I assume that if I did another square then I will get the answer of 8 because both grids have gave me an answer of 8, but just in case I will do a final two by two grid to prove that the diagonal difference is 8.
22 times 29 = 638
30 times 21= 630
Diagonal difference: 638 – 630 = 8
After doing this, I found out that the diagonal difference of two by two grids were eight because all the two by two grids gave me an answer of eight.
I will now find the diagonal difference for three by three grids.
I predict that because the diagonal difference of a two by two grid is eight, the diagonal difference for a three by three grid must be the following:
8 divided by 2 (to get diagonal difference of 1 by 1 grid) = 4
So 1 by 1 grid should equal to four
Three by three grids:
3 times 4 = 12
Three by three sized grids
I will now try to find the diagonal difference for three by three grids.
After working out all of the diagonal difference of all the different sizes, I will show this information below
2 by 2 grid = 8
3 by 3 grid = 32
4 by 4 grid = 72
5 by 5 grid = 128
6 by 6 grid = 200
7 by 7 grid = 392
I will put this information into an nth term table to make it easier for me to come up with a formula and it will be easier to see if any pattern emerges.
24 40 56 72 288 392
16 16 16 16 16
This tells me that because there is a second difference this is a quadratic equation. And because it is a quadratic equation, I will halve what I have found as the first part of my equation, which is 16: -
16 divided by 2 = 8
This tells me that eight is in my equation.
Therefore, eight is the first part of my equation.
I now have to find the n part in my equation to complete my equation; I will draw a graph to do this.
To get 8 from n=1 in the Nth term table, I have to times 8 times to get the first number which is 8, so n must be, n1, but it does not work for 32, so I will try n squared.
I will try it for the second number to see if it works,
8 times n squared
8 times 2 squared
8 times 4 = 32
This formula works, so I have now found the whole formula
The formula is!
8 times n squared
I will now define to explain what my formula means.
The formula means (defining n)
N= the size of the grid inside the eight by eight grid
8= the eight by eight grid
