2D cross shape - Rearranging the Shape

Then I only had 1 square left

Then I looked at each of the shapes I produced in their relation to n or what term the cross shape was in the sequence.

The cross shape I made my shapes from was 4th in the sequence the side of the squares were 3?3 and the length of the 2 rectangles were 3 all the measurements I had found were one less than n, all except from the 1 square by itself

An equation I could use for one of the squares was +(n-1)²

For one of the rectangles was +(n-1)

And for the single square by itself was +1

When I put this together I got (n-1)²+(n-)+1

But I had 2 squares and 2 rectangles so then I got 2(n-1)²+2(n-1)+1

Then I simplified this equation:

2(n-1)²+2(n-1)+1

2(n-1)(n-1)+2(n-1)+1

2(n-1)(n-1)+2n-2+1

2(n²-n-n+1)+2n-2+1

2n²-4n+2n+2-2+1

2n²-2n+1

I had simplified the equation as far as I could so now I could apply it to the shapes

On the next page I will show the table of results for my equation and show how they can be applied to the patterns of numbers

When I am given any cross shape to construct I am going to need to know how many of the squares are going to be white.

To find out this formula I will look ate the correlation between the white squares on a known shape and n

Each of the corners is getting used more than once because the lengths overlap which is why I get 4 more than I should

This shape is 4th in the series so n=4 because each side has a length of 4.

Also because the shape has 4 sides I now have 4n.

But there is on problem, the results I get are 4 more than the real totals.

This is because there are 4 corners and each side has 2 of them. So when I do 4n I am giving my amount for white squares 8 corners. On each one I will have to -4 to get the correct amount for white squares. My new equation for white squares is

4n-4

To find the number of black squares I have to take this equation from the total one.

Backwards

In the sequence I found that there are also previous squares in the equation.

I found them by reversing the equation. There were 2 previous ones