# My job is to investigate how many squares would be needed to make any cross shape like this build up in the same way.

Adeel Younis, Math’s Coursework Borders, Mr. Ekwalla, 10A, 10G 5/10/2007

Borders: Part 1

My job is to investigate how many squares would be needed to make any cross shape like this build up in the same way.

Below are diagrams of the cross shape pattern to the 8th sequence:

Here is my table of results, with the number of black, red and total squares

Now so I can find an equation which will tell me how many squares there will be in each sequence, I will find the differences for the black squares first.

To find the equation I will need bits of information, the equation Is an+b, where a is the difference and b is the 0th term, e.g. the first term here is 4, to find the 0th terms you need to go back one step (take the difference from the first terms to find the 0th term) and n is the term you want to find, e.g. if you want to find the 1st term n=1 if you want to find the second term n=2.

a=4, b=0. Now I have the equation, 4n+0, now let’s test it! We know the first terms is 4, so 4xn(1)+0, 4x1=4 so the answer is 4 and that is the 1st term.

Now ill do the same with the red squares

To find this equation I will need different bits of information, the equation is an2+bn+c,

A is ½ of the second difference, in this case a = 2

B is complicated, to find b you will need to find c first and substitute a and ...